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Properties

Label 1650.2.f.a.1649.3
Level $1650$
Weight $2$
Character 1650.1649
Analytic conductor $13.175$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1650,2,Mod(1649,1650)] 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("1650.1649"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1650.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.1753163335\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 66)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1649.3
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1650.1649
Dual form 1650.2.f.a.1649.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +(-1.41421 + 1.00000i) q^{3} -1.00000 q^{4} +(-1.00000 - 1.41421i) q^{6} -4.24264 q^{7} -1.00000i q^{8} +(1.00000 - 2.82843i) q^{9} +(-3.00000 + 1.41421i) q^{11} +(1.41421 - 1.00000i) q^{12} -4.24264 q^{13} -4.24264i q^{14} +1.00000 q^{16} +(2.82843 + 1.00000i) q^{18} +(6.00000 - 4.24264i) q^{21} +(-1.41421 - 3.00000i) q^{22} -1.41421 q^{23} +(1.00000 + 1.41421i) q^{24} -4.24264i q^{26} +(1.41421 + 5.00000i) q^{27} +4.24264 q^{28} +6.00000 q^{29} -4.00000 q^{31} +1.00000i q^{32} +(2.82843 - 5.00000i) q^{33} +(-1.00000 + 2.82843i) q^{36} +2.00000i q^{37} +(6.00000 - 4.24264i) q^{39} +6.00000 q^{41} +(4.24264 + 6.00000i) q^{42} -8.48528 q^{43} +(3.00000 - 1.41421i) q^{44} -1.41421i q^{46} +9.89949 q^{47} +(-1.41421 + 1.00000i) q^{48} +11.0000 q^{49} +4.24264 q^{52} +7.07107 q^{53} +(-5.00000 + 1.41421i) q^{54} +4.24264i q^{56} +6.00000i q^{58} -11.3137i q^{59} +4.24264i q^{61} -4.00000i q^{62} +(-4.24264 + 12.0000i) q^{63} -1.00000 q^{64} +(5.00000 + 2.82843i) q^{66} -4.00000i q^{67} +(2.00000 - 1.41421i) q^{69} +7.07107i q^{71} +(-2.82843 - 1.00000i) q^{72} -2.00000 q^{74} +(12.7279 - 6.00000i) q^{77} +(4.24264 + 6.00000i) q^{78} -4.24264i q^{79} +(-7.00000 - 5.65685i) q^{81} +6.00000i q^{82} -12.0000i q^{83} +(-6.00000 + 4.24264i) q^{84} -8.48528i q^{86} +(-8.48528 + 6.00000i) q^{87} +(1.41421 + 3.00000i) q^{88} +5.65685i q^{89} +18.0000 q^{91} +1.41421 q^{92} +(5.65685 - 4.00000i) q^{93} +9.89949i q^{94} +(-1.00000 - 1.41421i) q^{96} +8.00000i q^{97} +11.0000i q^{98} +(1.00000 + 9.89949i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6} + 4 q^{9} - 12 q^{11} + 4 q^{16} + 24 q^{21} + 4 q^{24} + 24 q^{29} - 16 q^{31} - 4 q^{36} + 24 q^{39} + 24 q^{41} + 12 q^{44} + 44 q^{49} - 20 q^{54} - 4 q^{64} + 20 q^{66} + 8 q^{69}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1650\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(727\) \(1201\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(3\) −1.41421 + 1.00000i −0.816497 + 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 − 1.41421i −0.408248 − 0.577350i
\(7\) −4.24264 −1.60357 −0.801784 − 0.597614i \(-0.796115\pi\)
−0.801784 + 0.597614i \(0.796115\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 − 2.82843i 0.333333 − 0.942809i
\(10\) 0 0
\(11\) −3.00000 + 1.41421i −0.904534 + 0.426401i
\(12\) 1.41421 − 1.00000i 0.408248 − 0.288675i
\(13\) −4.24264 −1.17670 −0.588348 − 0.808608i \(-0.700222\pi\)
−0.588348 + 0.808608i \(0.700222\pi\)
\(14\) − 4.24264i − 1.13389i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 2.82843 + 1.00000i 0.666667 + 0.235702i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 6.00000 − 4.24264i 1.30931 − 0.925820i
\(22\) −1.41421 − 3.00000i −0.301511 − 0.639602i
\(23\) −1.41421 −0.294884 −0.147442 − 0.989071i \(-0.547104\pi\)
−0.147442 + 0.989071i \(0.547104\pi\)
\(24\) 1.00000 + 1.41421i 0.204124 + 0.288675i
\(25\) 0 0
\(26\) − 4.24264i − 0.832050i
\(27\) 1.41421 + 5.00000i 0.272166 + 0.962250i
\(28\) 4.24264 0.801784
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 2.82843 − 5.00000i 0.492366 − 0.870388i
\(34\) 0 0
\(35\) 0 0
\(36\) −1.00000 + 2.82843i −0.166667 + 0.471405i
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 6.00000 − 4.24264i 0.960769 − 0.679366i
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 4.24264 + 6.00000i 0.654654 + 0.925820i
\(43\) −8.48528 −1.29399 −0.646997 − 0.762493i \(-0.723975\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 3.00000 − 1.41421i 0.452267 − 0.213201i
\(45\) 0 0
\(46\) − 1.41421i − 0.208514i
\(47\) 9.89949 1.44399 0.721995 − 0.691898i \(-0.243225\pi\)
0.721995 + 0.691898i \(0.243225\pi\)
\(48\) −1.41421 + 1.00000i −0.204124 + 0.144338i
\(49\) 11.0000 1.57143
\(50\) 0 0
\(51\) 0 0
\(52\) 4.24264 0.588348
\(53\) 7.07107 0.971286 0.485643 − 0.874157i \(-0.338586\pi\)
0.485643 + 0.874157i \(0.338586\pi\)
\(54\) −5.00000 + 1.41421i −0.680414 + 0.192450i
\(55\) 0 0
\(56\) 4.24264i 0.566947i
\(57\) 0 0
\(58\) 6.00000i 0.787839i
\(59\) − 11.3137i − 1.47292i −0.676481 − 0.736460i \(-0.736496\pi\)
0.676481 − 0.736460i \(-0.263504\pi\)
\(60\) 0 0
\(61\) 4.24264i 0.543214i 0.962408 + 0.271607i \(0.0875552\pi\)
−0.962408 + 0.271607i \(0.912445\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) −4.24264 + 12.0000i −0.534522 + 1.51186i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 5.00000 + 2.82843i 0.615457 + 0.348155i
\(67\) − 4.00000i − 0.488678i −0.969690 − 0.244339i \(-0.921429\pi\)
0.969690 − 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) 2.00000 − 1.41421i 0.240772 − 0.170251i
\(70\) 0 0
\(71\) 7.07107i 0.839181i 0.907713 + 0.419591i \(0.137826\pi\)
−0.907713 + 0.419591i \(0.862174\pi\)
\(72\) −2.82843 − 1.00000i −0.333333 − 0.117851i
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 0 0
\(77\) 12.7279 − 6.00000i 1.45048 − 0.683763i
\(78\) 4.24264 + 6.00000i 0.480384 + 0.679366i
\(79\) − 4.24264i − 0.477334i −0.971101 − 0.238667i \(-0.923290\pi\)
0.971101 − 0.238667i \(-0.0767105\pi\)
\(80\) 0 0
\(81\) −7.00000 − 5.65685i −0.777778 − 0.628539i
\(82\) 6.00000i 0.662589i
\(83\) − 12.0000i − 1.31717i −0.752506 − 0.658586i \(-0.771155\pi\)
0.752506 − 0.658586i \(-0.228845\pi\)
\(84\) −6.00000 + 4.24264i −0.654654 + 0.462910i
\(85\) 0 0
\(86\) − 8.48528i − 0.914991i
\(87\) −8.48528 + 6.00000i −0.909718 + 0.643268i
\(88\) 1.41421 + 3.00000i 0.150756 + 0.319801i
\(89\) 5.65685i 0.599625i 0.953998 + 0.299813i \(0.0969242\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) 18.0000 1.88691
\(92\) 1.41421 0.147442
\(93\) 5.65685 − 4.00000i 0.586588 − 0.414781i
\(94\) 9.89949i 1.02105i
\(95\) 0 0
\(96\) −1.00000 − 1.41421i −0.102062 − 0.144338i
\(97\) 8.00000i 0.812277i 0.913812 + 0.406138i \(0.133125\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 11.0000i 1.11117i
\(99\) 1.00000 + 9.89949i 0.100504 + 0.994937i
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 1650.2.f.a.1649.3 4
3.2 odd 2 1650.2.f.b.1649.2 4
5.2 odd 4 1650.2.d.a.1451.2 2
5.3 odd 4 66.2.b.b.65.1 yes 2
5.4 even 2 inner 1650.2.f.a.1649.2 4
11.10 odd 2 1650.2.f.b.1649.1 4
15.2 even 4 1650.2.d.b.1451.1 2
15.8 even 4 66.2.b.a.65.2 yes 2
15.14 odd 2 1650.2.f.b.1649.3 4
20.3 even 4 528.2.b.c.65.2 2
33.32 even 2 inner 1650.2.f.a.1649.4 4
40.3 even 4 2112.2.b.b.65.1 2
40.13 odd 4 2112.2.b.i.65.2 2
45.13 odd 12 1782.2.i.c.593.1 4
45.23 even 12 1782.2.i.f.593.2 4
45.38 even 12 1782.2.i.f.1187.1 4
45.43 odd 12 1782.2.i.c.1187.2 4
55.3 odd 20 726.2.h.e.233.2 8
55.8 even 20 726.2.h.i.233.2 8
55.13 even 20 726.2.h.i.161.1 8
55.18 even 20 726.2.h.i.215.1 8
55.28 even 20 726.2.h.i.239.2 8
55.32 even 4 1650.2.d.b.1451.2 2
55.38 odd 20 726.2.h.e.239.2 8
55.43 even 4 66.2.b.a.65.1 ✓ 2
55.48 odd 20 726.2.h.e.215.1 8
55.53 odd 20 726.2.h.e.161.1 8
55.54 odd 2 1650.2.f.b.1649.4 4
60.23 odd 4 528.2.b.b.65.1 2
120.53 even 4 2112.2.b.g.65.1 2
120.83 odd 4 2112.2.b.d.65.2 2
165.8 odd 20 726.2.h.e.233.1 8
165.32 odd 4 1650.2.d.a.1451.1 2
165.38 even 20 726.2.h.i.239.1 8
165.53 even 20 726.2.h.i.161.2 8
165.68 odd 20 726.2.h.e.161.2 8
165.83 odd 20 726.2.h.e.239.1 8
165.98 odd 4 66.2.b.b.65.2 yes 2
165.113 even 20 726.2.h.i.233.1 8
165.128 odd 20 726.2.h.e.215.2 8
165.158 even 20 726.2.h.i.215.2 8
165.164 even 2 inner 1650.2.f.a.1649.1 4
220.43 odd 4 528.2.b.b.65.2 2
440.43 odd 4 2112.2.b.d.65.1 2
440.373 even 4 2112.2.b.g.65.2 2
495.43 even 12 1782.2.i.f.1187.2 4
495.263 odd 12 1782.2.i.c.1187.1 4
495.373 even 12 1782.2.i.f.593.1 4
495.428 odd 12 1782.2.i.c.593.2 4
660.263 even 4 528.2.b.c.65.1 2
1320.923 even 4 2112.2.b.b.65.2 2
1320.1253 odd 4 2112.2.b.i.65.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
66.2.b.a.65.1 ✓ 2 55.43 even 4
66.2.b.a.65.2 yes 2 15.8 even 4
66.2.b.b.65.1 yes 2 5.3 odd 4
66.2.b.b.65.2 yes 2 165.98 odd 4
528.2.b.b.65.1 2 60.23 odd 4
528.2.b.b.65.2 2 220.43 odd 4
528.2.b.c.65.1 2 660.263 even 4
528.2.b.c.65.2 2 20.3 even 4
726.2.h.e.161.1 8 55.53 odd 20
726.2.h.e.161.2 8 165.68 odd 20
726.2.h.e.215.1 8 55.48 odd 20
726.2.h.e.215.2 8 165.128 odd 20
726.2.h.e.233.1 8 165.8 odd 20
726.2.h.e.233.2 8 55.3 odd 20
726.2.h.e.239.1 8 165.83 odd 20
726.2.h.e.239.2 8 55.38 odd 20
726.2.h.i.161.1 8 55.13 even 20
726.2.h.i.161.2 8 165.53 even 20
726.2.h.i.215.1 8 55.18 even 20
726.2.h.i.215.2 8 165.158 even 20
726.2.h.i.233.1 8 165.113 even 20
726.2.h.i.233.2 8 55.8 even 20
726.2.h.i.239.1 8 165.38 even 20
726.2.h.i.239.2 8 55.28 even 20
1650.2.d.a.1451.1 2 165.32 odd 4
1650.2.d.a.1451.2 2 5.2 odd 4
1650.2.d.b.1451.1 2 15.2 even 4
1650.2.d.b.1451.2 2 55.32 even 4
1650.2.f.a.1649.1 4 165.164 even 2 inner
1650.2.f.a.1649.2 4 5.4 even 2 inner
1650.2.f.a.1649.3 4 1.1 even 1 trivial
1650.2.f.a.1649.4 4 33.32 even 2 inner
1650.2.f.b.1649.1 4 11.10 odd 2
1650.2.f.b.1649.2 4 3.2 odd 2
1650.2.f.b.1649.3 4 15.14 odd 2
1650.2.f.b.1649.4 4 55.54 odd 2
1782.2.i.c.593.1 4 45.13 odd 12
1782.2.i.c.593.2 4 495.428 odd 12
1782.2.i.c.1187.1 4 495.263 odd 12
1782.2.i.c.1187.2 4 45.43 odd 12
1782.2.i.f.593.1 4 495.373 even 12
1782.2.i.f.593.2 4 45.23 even 12
1782.2.i.f.1187.1 4 45.38 even 12
1782.2.i.f.1187.2 4 495.43 even 12
2112.2.b.b.65.1 2 40.3 even 4
2112.2.b.b.65.2 2 1320.923 even 4
2112.2.b.d.65.1 2 440.43 odd 4
2112.2.b.d.65.2 2 120.83 odd 4
2112.2.b.g.65.1 2 120.53 even 4
2112.2.b.g.65.2 2 440.373 even 4
2112.2.b.i.65.1 2 1320.1253 odd 4
2112.2.b.i.65.2 2 40.13 odd 4