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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1450,2,Mod(1449,1450)] 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("1450.1449"); 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([1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1450.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == 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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1449.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1450.1449
Dual form 1450.2.d.c.1449.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.00000i q^{7} +1.00000 q^{8} -2.00000 q^{9} +5.00000i q^{11} -1.00000 q^{12} +1.00000i q^{13} -2.00000i q^{14} +1.00000 q^{16} -2.00000 q^{17} -2.00000 q^{18} +4.00000i q^{19} +2.00000i q^{21} +5.00000i q^{22} +6.00000i q^{23} -1.00000 q^{24} +1.00000i q^{26} +5.00000 q^{27} -2.00000i q^{28} +(-5.00000 + 2.00000i) q^{29} -5.00000i q^{31} +1.00000 q^{32} -5.00000i q^{33} -2.00000 q^{34} -2.00000 q^{36} +8.00000 q^{37} +4.00000i q^{38} -1.00000i q^{39} +10.0000i q^{41} +2.00000i q^{42} +9.00000 q^{43} +5.00000i q^{44} +6.00000i q^{46} +3.00000 q^{47} -1.00000 q^{48} +3.00000 q^{49} +2.00000 q^{51} +1.00000i q^{52} +1.00000i q^{53} +5.00000 q^{54} -2.00000i q^{56} -4.00000i q^{57} +(-5.00000 + 2.00000i) q^{58} -10.0000 q^{59} +10.0000i q^{61} -5.00000i q^{62} +4.00000i q^{63} +1.00000 q^{64} -5.00000i q^{66} +8.00000i q^{67} -2.00000 q^{68} -6.00000i q^{69} -8.00000 q^{71} -2.00000 q^{72} -16.0000 q^{73} +8.00000 q^{74} +4.00000i q^{76} +10.0000 q^{77} -1.00000i q^{78} -1.00000i q^{79} +1.00000 q^{81} +10.0000i q^{82} -14.0000i q^{83} +2.00000i q^{84} +9.00000 q^{86} +(5.00000 - 2.00000i) q^{87} +5.00000i q^{88} +14.0000i q^{89} +2.00000 q^{91} +6.00000i q^{92} +5.00000i q^{93} +3.00000 q^{94} -1.00000 q^{96} -2.00000 q^{97} +3.00000 q^{98} -10.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{8} - 4 q^{9} - 2 q^{12} + 2 q^{16} - 4 q^{17} - 4 q^{18} - 2 q^{24} + 10 q^{27} - 10 q^{29} + 2 q^{32} - 4 q^{34} - 4 q^{36} + 16 q^{37} + 18 q^{43} + 6 q^{47}+ \cdots + 6 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) − 2.00000i − 0.755929i −0.925820 − 0.377964i \(-0.876624\pi\)
0.925820 − 0.377964i \(-0.123376\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 5.00000i 1.50756i 0.657129 + 0.753778i \(0.271771\pi\)
−0.657129 + 0.753778i \(0.728229\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000i 0.277350i 0.990338 + 0.138675i \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) − 2.00000i − 0.534522i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −2.00000 −0.471405
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) 2.00000i 0.436436i
\(22\) 5.00000i 1.06600i
\(23\) 6.00000i 1.25109i 0.780189 + 0.625543i \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 1.00000i 0.196116i
\(27\) 5.00000 0.962250
\(28\) − 2.00000i − 0.377964i
\(29\) −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 \(-0.148224\pi\)
\(32\) 1.00000 0.176777
\(33\) − 5.00000i − 0.870388i
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 4.00000i 0.648886i
\(39\) − 1.00000i − 0.160128i
\(40\) 0 0
\(41\) 10.0000i 1.56174i 0.624695 + 0.780869i \(0.285223\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 2.00000i 0.308607i
\(43\) 9.00000 1.37249 0.686244 − 0.727372i \(-0.259258\pi\)
0.686244 + 0.727372i \(0.259258\pi\)
\(44\) 5.00000i 0.753778i
\(45\) 0 0
\(46\) 6.00000i 0.884652i
\(47\) 3.00000 0.437595 0.218797 − 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) −1.00000 −0.144338
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 1.00000i 0.138675i
\(53\) 1.00000i 0.137361i 0.997639 + 0.0686803i \(0.0218788\pi\)
−0.997639 + 0.0686803i \(0.978121\pi\)
\(54\) 5.00000 0.680414
\(55\) 0 0
\(56\) − 2.00000i − 0.267261i
\(57\) − 4.00000i − 0.529813i
\(58\) −5.00000 + 2.00000i −0.656532 + 0.262613i
\(59\) −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 \(0.221142\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) − 5.00000i − 0.635001i
\(63\) 4.00000i 0.503953i
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) − 5.00000i − 0.615457i
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) −2.00000 −0.242536
\(69\) − 6.00000i − 0.722315i
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −2.00000 −0.235702
\(73\) −16.0000 −1.87266 −0.936329 − 0.351123i \(-0.885800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) 4.00000i 0.458831i
\(77\) 10.0000 1.13961
\(78\) − 1.00000i − 0.113228i
\(79\) − 1.00000i − 0.112509i −0.998416 − 0.0562544i \(-0.982084\pi\)
0.998416 − 0.0562544i \(-0.0179158\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 10.0000i 1.10432i
\(83\) − 14.0000i − 1.53670i −0.640030 − 0.768350i \(-0.721078\pi\)
0.640030 − 0.768350i \(-0.278922\pi\)
\(84\) 2.00000i 0.218218i
\(85\) 0 0
\(86\) 9.00000 0.970495
\(87\) 5.00000 − 2.00000i 0.536056 − 0.214423i
\(88\) 5.00000i 0.533002i
\(89\) 14.0000i 1.48400i 0.670402 + 0.741999i \(0.266122\pi\)
−0.670402 + 0.741999i \(0.733878\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 6.00000i 0.625543i
\(93\) 5.00000i 0.518476i
\(94\) 3.00000 0.309426
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 3.00000 0.303046
\(99\) − 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.d.c.1449.1 2
5.2 odd 4 1450.2.c.a.1101.2 2
5.3 odd 4 58.2.b.a.57.1 ✓ 2
5.4 even 2 1450.2.d.b.1449.2 2
15.8 even 4 522.2.d.a.289.2 2
20.3 even 4 464.2.e.c.289.2 2
29.28 even 2 1450.2.d.b.1449.1 2
40.3 even 4 1856.2.e.d.1217.1 2
40.13 odd 4 1856.2.e.b.1217.2 2
60.23 odd 4 4176.2.o.d.289.2 2
145.28 odd 4 58.2.b.a.57.2 yes 2
145.57 odd 4 1450.2.c.a.1101.1 2
145.128 even 4 1682.2.a.g.1.1 1
145.133 even 4 1682.2.a.c.1.1 1
145.144 even 2 inner 1450.2.d.c.1449.2 2
435.173 even 4 522.2.d.a.289.1 2
580.463 even 4 464.2.e.c.289.1 2
1160.173 odd 4 1856.2.e.b.1217.1 2
1160.1043 even 4 1856.2.e.d.1217.2 2
1740.1043 odd 4 4176.2.o.d.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.b.a.57.1 ✓ 2 5.3 odd 4
58.2.b.a.57.2 yes 2 145.28 odd 4
464.2.e.c.289.1 2 580.463 even 4
464.2.e.c.289.2 2 20.3 even 4
522.2.d.a.289.1 2 435.173 even 4
522.2.d.a.289.2 2 15.8 even 4
1450.2.c.a.1101.1 2 145.57 odd 4
1450.2.c.a.1101.2 2 5.2 odd 4
1450.2.d.b.1449.1 2 29.28 even 2
1450.2.d.b.1449.2 2 5.4 even 2
1450.2.d.c.1449.1 2 1.1 even 1 trivial
1450.2.d.c.1449.2 2 145.144 even 2 inner
1682.2.a.c.1.1 1 145.133 even 4
1682.2.a.g.1.1 1 145.128 even 4
1856.2.e.b.1217.1 2 1160.173 odd 4
1856.2.e.b.1217.2 2 40.13 odd 4
1856.2.e.d.1217.1 2 40.3 even 4
1856.2.e.d.1217.2 2 1160.1043 even 4
4176.2.o.d.289.1 2 1740.1043 odd 4
4176.2.o.d.289.2 2 60.23 odd 4