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Properties

Label 1050.2.o.b.949.1
Level $1050$
Weight $2$
Character 1050.949
Analytic conductor $8.384$
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] = [1050,2,Mod(499,1050)] 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("1050.499"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1050, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,2,0,-4,0,0,2,0,-6,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 949.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1050.949
Dual form 1050.2.o.b.499.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 0.500000i) q^{2} +(0.866025 + 0.500000i) q^{3} +(0.500000 - 0.866025i) q^{4} -1.00000 q^{6} +(0.866025 + 2.50000i) q^{7} +1.00000i q^{8} +(0.500000 + 0.866025i) q^{9} +(-1.50000 + 2.59808i) q^{11} +(0.866025 - 0.500000i) q^{12} +4.00000i q^{13} +(-2.00000 - 1.73205i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-0.866025 - 0.500000i) q^{18} +(-2.00000 - 3.46410i) q^{19} +(-0.500000 + 2.59808i) q^{21} -3.00000i q^{22} +(-0.500000 + 0.866025i) q^{24} +(-2.00000 - 3.46410i) q^{26} +1.00000i q^{27} +(2.59808 + 0.500000i) q^{28} -9.00000 q^{29} +(0.500000 - 0.866025i) q^{31} +(0.866025 + 0.500000i) q^{32} +(-2.59808 + 1.50000i) q^{33} +1.00000 q^{36} +(6.92820 - 4.00000i) q^{37} +(3.46410 + 2.00000i) q^{38} +(-2.00000 + 3.46410i) q^{39} +(-0.866025 - 2.50000i) q^{42} +10.0000i q^{43} +(1.50000 + 2.59808i) q^{44} +(-5.19615 + 3.00000i) q^{47} -1.00000i q^{48} +(-5.50000 + 4.33013i) q^{49} +(3.46410 + 2.00000i) q^{52} +(-2.59808 - 1.50000i) q^{53} +(-0.500000 - 0.866025i) q^{54} +(-2.50000 + 0.866025i) q^{56} -4.00000i q^{57} +(7.79423 - 4.50000i) q^{58} +(1.50000 - 2.59808i) q^{59} +(5.00000 + 8.66025i) q^{61} +1.00000i q^{62} +(-1.73205 + 2.00000i) q^{63} -1.00000 q^{64} +(1.50000 - 2.59808i) q^{66} +(8.66025 + 5.00000i) q^{67} -6.00000 q^{71} +(-0.866025 + 0.500000i) q^{72} +(1.73205 + 1.00000i) q^{73} +(-4.00000 + 6.92820i) q^{74} -4.00000 q^{76} +(-7.79423 - 1.50000i) q^{77} -4.00000i q^{78} +(-0.500000 - 0.866025i) q^{79} +(-0.500000 + 0.866025i) q^{81} +9.00000i q^{83} +(2.00000 + 1.73205i) q^{84} +(-5.00000 - 8.66025i) q^{86} +(-7.79423 - 4.50000i) q^{87} +(-2.59808 - 1.50000i) q^{88} +(3.00000 + 5.19615i) q^{89} +(-10.0000 + 3.46410i) q^{91} +(0.866025 - 0.500000i) q^{93} +(3.00000 - 5.19615i) q^{94} +(0.500000 + 0.866025i) q^{96} -1.00000i q^{97} +(2.59808 - 6.50000i) q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 4 q^{6} + 2 q^{9} - 6 q^{11} - 8 q^{14} - 2 q^{16} - 8 q^{19} - 2 q^{21} - 2 q^{24} - 8 q^{26} - 36 q^{29} + 2 q^{31} + 4 q^{36} - 8 q^{39} + 6 q^{44} - 22 q^{49} - 2 q^{54} - 10 q^{56}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 + 0.500000i −0.612372 + 0.353553i
\(3\) 0.866025 + 0.500000i 0.500000 + 0.288675i
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 0.866025 + 2.50000i 0.327327 + 0.944911i
\(8\) 1.00000i 0.353553i
\(9\) 0.500000 + 0.866025i 0.166667 + 0.288675i
\(10\) 0 0
\(11\) −1.50000 + 2.59808i −0.452267 + 0.783349i −0.998526 − 0.0542666i \(-0.982718\pi\)
0.546259 + 0.837616i \(0.316051\pi\)
\(12\) 0.866025 − 0.500000i 0.250000 − 0.144338i
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −2.00000 − 1.73205i −0.534522 − 0.462910i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) −0.866025 − 0.500000i −0.204124 − 0.117851i
\(19\) −2.00000 − 3.46410i −0.458831 − 0.794719i 0.540068 − 0.841621i \(-0.318398\pi\)
−0.998899 + 0.0469020i \(0.985065\pi\)
\(20\) 0 0
\(21\) −0.500000 + 2.59808i −0.109109 + 0.566947i
\(22\) − 3.00000i − 0.639602i
\(23\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) −0.500000 + 0.866025i −0.102062 + 0.176777i
\(25\) 0 0
\(26\) −2.00000 − 3.46410i −0.392232 − 0.679366i
\(27\) 1.00000i 0.192450i
\(28\) 2.59808 + 0.500000i 0.490990 + 0.0944911i
\(29\) −9.00000 −1.67126 −0.835629 − 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) 0.500000 − 0.866025i 0.0898027 − 0.155543i −0.817625 − 0.575751i \(-0.804710\pi\)
0.907428 + 0.420208i \(0.138043\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) −2.59808 + 1.50000i −0.452267 + 0.261116i
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.92820 − 4.00000i 1.13899 − 0.657596i 0.192809 − 0.981236i \(-0.438240\pi\)
0.946180 + 0.323640i \(0.104907\pi\)
\(38\) 3.46410 + 2.00000i 0.561951 + 0.324443i
\(39\) −2.00000 + 3.46410i −0.320256 + 0.554700i
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −0.866025 − 2.50000i −0.133631 − 0.385758i
\(43\) 10.0000i 1.52499i 0.646997 + 0.762493i \(0.276025\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 1.50000 + 2.59808i 0.226134 + 0.391675i
\(45\) 0 0
\(46\) 0 0
\(47\) −5.19615 + 3.00000i −0.757937 + 0.437595i −0.828554 − 0.559908i \(-0.810836\pi\)
0.0706177 + 0.997503i \(0.477503\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −5.50000 + 4.33013i −0.785714 + 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) 3.46410 + 2.00000i 0.480384 + 0.277350i
\(53\) −2.59808 − 1.50000i −0.356873 − 0.206041i 0.310835 − 0.950464i \(-0.399391\pi\)
−0.667708 + 0.744423i \(0.732725\pi\)
\(54\) −0.500000 − 0.866025i −0.0680414 − 0.117851i
\(55\) 0 0
\(56\) −2.50000 + 0.866025i −0.334077 + 0.115728i
\(57\) − 4.00000i − 0.529813i
\(58\) 7.79423 − 4.50000i 1.02343 − 0.590879i
\(59\) 1.50000 − 2.59808i 0.195283 − 0.338241i −0.751710 − 0.659494i \(-0.770771\pi\)
0.946993 + 0.321253i \(0.104104\pi\)
\(60\) 0 0
\(61\) 5.00000 + 8.66025i 0.640184 + 1.10883i 0.985391 + 0.170305i \(0.0544754\pi\)
−0.345207 + 0.938527i \(0.612191\pi\)
\(62\) 1.00000i 0.127000i
\(63\) −1.73205 + 2.00000i −0.218218 + 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 1.50000 − 2.59808i 0.184637 − 0.319801i
\(67\) 8.66025 + 5.00000i 1.05802 + 0.610847i 0.924883 − 0.380251i \(-0.124162\pi\)
0.133135 + 0.991098i \(0.457496\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) −0.866025 + 0.500000i −0.102062 + 0.0589256i
\(73\) 1.73205 + 1.00000i 0.202721 + 0.117041i 0.597924 − 0.801553i \(-0.295992\pi\)
−0.395203 + 0.918594i \(0.629326\pi\)
\(74\) −4.00000 + 6.92820i −0.464991 + 0.805387i
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) −7.79423 − 1.50000i −0.888235 − 0.170941i
\(78\) − 4.00000i − 0.452911i
\(79\) −0.500000 − 0.866025i −0.0562544 − 0.0974355i 0.836527 − 0.547926i \(-0.184582\pi\)
−0.892781 + 0.450490i \(0.851249\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 9.00000i 0.987878i 0.869496 + 0.493939i \(0.164443\pi\)
−0.869496 + 0.493939i \(0.835557\pi\)
\(84\) 2.00000 + 1.73205i 0.218218 + 0.188982i
\(85\) 0 0
\(86\) −5.00000 − 8.66025i −0.539164 − 0.933859i
\(87\) −7.79423 − 4.50000i −0.835629 − 0.482451i
\(88\) −2.59808 − 1.50000i −0.276956 − 0.159901i
\(89\) 3.00000 + 5.19615i 0.317999 + 0.550791i 0.980071 − 0.198650i \(-0.0636557\pi\)
−0.662071 + 0.749441i \(0.730322\pi\)
\(90\) 0 0
\(91\) −10.0000 + 3.46410i −1.04828 + 0.363137i
\(92\) 0 0
\(93\) 0.866025 − 0.500000i 0.0898027 − 0.0518476i
\(94\) 3.00000 − 5.19615i 0.309426 − 0.535942i
\(95\) 0 0
\(96\) 0.500000 + 0.866025i 0.0510310 + 0.0883883i
\(97\) − 1.00000i − 0.101535i −0.998711 − 0.0507673i \(-0.983833\pi\)
0.998711 − 0.0507673i \(-0.0161667\pi\)
\(98\) 2.59808 − 6.50000i 0.262445 − 0.656599i
\(99\) −3.00000 −0.301511
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 1050.2.o.b.949.1 4
5.2 odd 4 1050.2.i.e.151.1 2
5.3 odd 4 42.2.e.b.25.1 ✓ 2
5.4 even 2 inner 1050.2.o.b.949.2 4
7.2 even 3 inner 1050.2.o.b.499.2 4
15.8 even 4 126.2.g.b.109.1 2
20.3 even 4 336.2.q.d.193.1 2
35.2 odd 12 1050.2.i.e.751.1 2
35.3 even 12 294.2.a.a.1.1 1
35.9 even 6 inner 1050.2.o.b.499.1 4
35.13 even 4 294.2.e.f.67.1 2
35.17 even 12 7350.2.a.cw.1.1 1
35.18 odd 12 294.2.a.d.1.1 1
35.23 odd 12 42.2.e.b.37.1 yes 2
35.32 odd 12 7350.2.a.ce.1.1 1
35.33 even 12 294.2.e.f.79.1 2
40.3 even 4 1344.2.q.j.193.1 2
40.13 odd 4 1344.2.q.v.193.1 2
45.13 odd 12 1134.2.e.a.865.1 2
45.23 even 12 1134.2.e.p.865.1 2
45.38 even 12 1134.2.h.a.109.1 2
45.43 odd 12 1134.2.h.p.109.1 2
60.23 odd 4 1008.2.s.n.865.1 2
105.23 even 12 126.2.g.b.37.1 2
105.38 odd 12 882.2.a.k.1.1 1
105.53 even 12 882.2.a.g.1.1 1
105.68 odd 12 882.2.g.b.667.1 2
105.83 odd 4 882.2.g.b.361.1 2
140.3 odd 12 2352.2.a.n.1.1 1
140.23 even 12 336.2.q.d.289.1 2
140.83 odd 4 2352.2.q.m.1537.1 2
140.103 odd 12 2352.2.q.m.961.1 2
140.123 even 12 2352.2.a.m.1.1 1
280.3 odd 12 9408.2.a.bm.1.1 1
280.53 odd 12 9408.2.a.d.1.1 1
280.93 odd 12 1344.2.q.v.961.1 2
280.123 even 12 9408.2.a.bu.1.1 1
280.163 even 12 1344.2.q.j.961.1 2
280.213 even 12 9408.2.a.db.1.1 1
315.23 even 12 1134.2.h.a.541.1 2
315.58 odd 12 1134.2.h.p.541.1 2
315.128 even 12 1134.2.e.p.919.1 2
315.268 odd 12 1134.2.e.a.919.1 2
420.23 odd 12 1008.2.s.n.289.1 2
420.143 even 12 7056.2.a.bz.1.1 1
420.263 odd 12 7056.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.e.b.25.1 ✓ 2 5.3 odd 4
42.2.e.b.37.1 yes 2 35.23 odd 12
126.2.g.b.37.1 2 105.23 even 12
126.2.g.b.109.1 2 15.8 even 4
294.2.a.a.1.1 1 35.3 even 12
294.2.a.d.1.1 1 35.18 odd 12
294.2.e.f.67.1 2 35.13 even 4
294.2.e.f.79.1 2 35.33 even 12
336.2.q.d.193.1 2 20.3 even 4
336.2.q.d.289.1 2 140.23 even 12
882.2.a.g.1.1 1 105.53 even 12
882.2.a.k.1.1 1 105.38 odd 12
882.2.g.b.361.1 2 105.83 odd 4
882.2.g.b.667.1 2 105.68 odd 12
1008.2.s.n.289.1 2 420.23 odd 12
1008.2.s.n.865.1 2 60.23 odd 4
1050.2.i.e.151.1 2 5.2 odd 4
1050.2.i.e.751.1 2 35.2 odd 12
1050.2.o.b.499.1 4 35.9 even 6 inner
1050.2.o.b.499.2 4 7.2 even 3 inner
1050.2.o.b.949.1 4 1.1 even 1 trivial
1050.2.o.b.949.2 4 5.4 even 2 inner
1134.2.e.a.865.1 2 45.13 odd 12
1134.2.e.a.919.1 2 315.268 odd 12
1134.2.e.p.865.1 2 45.23 even 12
1134.2.e.p.919.1 2 315.128 even 12
1134.2.h.a.109.1 2 45.38 even 12
1134.2.h.a.541.1 2 315.23 even 12
1134.2.h.p.109.1 2 45.43 odd 12
1134.2.h.p.541.1 2 315.58 odd 12
1344.2.q.j.193.1 2 40.3 even 4
1344.2.q.j.961.1 2 280.163 even 12
1344.2.q.v.193.1 2 40.13 odd 4
1344.2.q.v.961.1 2 280.93 odd 12
2352.2.a.m.1.1 1 140.123 even 12
2352.2.a.n.1.1 1 140.3 odd 12
2352.2.q.m.961.1 2 140.103 odd 12
2352.2.q.m.1537.1 2 140.83 odd 4
7056.2.a.g.1.1 1 420.263 odd 12
7056.2.a.bz.1.1 1 420.143 even 12
7350.2.a.ce.1.1 1 35.32 odd 12
7350.2.a.cw.1.1 1 35.17 even 12
9408.2.a.d.1.1 1 280.53 odd 12
9408.2.a.bm.1.1 1 280.3 odd 12
9408.2.a.bu.1.1 1 280.123 even 12
9408.2.a.db.1.1 1 280.213 even 12