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Properties

Label 1050.2.o.g.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,-4,0,0,2,0,10,0,0,2] 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 210)
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.g.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} +(-1.73205 + 2.00000i) q^{7} +1.00000i q^{8} +(0.500000 + 0.866025i) q^{9} +(2.50000 - 4.33013i) q^{11} +(0.866025 - 0.500000i) q^{12} -5.00000i q^{13} +(0.500000 - 2.59808i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-3.46410 - 2.00000i) q^{17} +(-0.866025 - 0.500000i) q^{18} +(-3.50000 - 6.06218i) q^{19} +(-2.50000 + 0.866025i) q^{21} +5.00000i q^{22} +(0.866025 - 0.500000i) q^{23} +(-0.500000 + 0.866025i) q^{24} +(2.50000 + 4.33013i) q^{26} +1.00000i q^{27} +(0.866025 + 2.50000i) q^{28} +(1.00000 - 1.73205i) q^{31} +(0.866025 + 0.500000i) q^{32} +(4.33013 - 2.50000i) q^{33} +4.00000 q^{34} +1.00000 q^{36} +(-0.866025 + 0.500000i) q^{37} +(6.06218 + 3.50000i) q^{38} +(2.50000 - 4.33013i) q^{39} +5.00000 q^{41} +(1.73205 - 2.00000i) q^{42} +12.0000i q^{43} +(-2.50000 - 4.33013i) q^{44} +(-0.500000 + 0.866025i) q^{46} +(9.52628 - 5.50000i) q^{47} -1.00000i q^{48} +(-1.00000 - 6.92820i) q^{49} +(-2.00000 - 3.46410i) q^{51} +(-4.33013 - 2.50000i) q^{52} +(7.79423 + 4.50000i) q^{53} +(-0.500000 - 0.866025i) q^{54} +(-2.00000 - 1.73205i) q^{56} -7.00000i q^{57} +(2.00000 - 3.46410i) q^{59} +(-2.00000 - 3.46410i) q^{61} +2.00000i q^{62} +(-2.59808 - 0.500000i) q^{63} -1.00000 q^{64} +(-2.50000 + 4.33013i) q^{66} +(-10.3923 - 6.00000i) q^{67} +(-3.46410 + 2.00000i) q^{68} +1.00000 q^{69} +2.00000 q^{71} +(-0.866025 + 0.500000i) q^{72} +(-8.66025 - 5.00000i) q^{73} +(0.500000 - 0.866025i) q^{74} -7.00000 q^{76} +(4.33013 + 12.5000i) q^{77} +5.00000i q^{78} +(-6.00000 - 10.3923i) q^{79} +(-0.500000 + 0.866025i) q^{81} +(-4.33013 + 2.50000i) q^{82} -12.0000i q^{83} +(-0.500000 + 2.59808i) q^{84} +(-6.00000 - 10.3923i) q^{86} +(4.33013 + 2.50000i) q^{88} +(7.00000 + 12.1244i) q^{89} +(10.0000 + 8.66025i) q^{91} -1.00000i q^{92} +(1.73205 - 1.00000i) q^{93} +(-5.50000 + 9.52628i) q^{94} +(0.500000 + 0.866025i) q^{96} +8.00000i q^{97} +(4.33013 + 5.50000i) q^{98} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 4 q^{6} + 2 q^{9} + 10 q^{11} + 2 q^{14} - 2 q^{16} - 14 q^{19} - 10 q^{21} - 2 q^{24} + 10 q^{26} + 4 q^{31} + 16 q^{34} + 4 q^{36} + 10 q^{39} + 20 q^{41} - 10 q^{44} - 2 q^{46} - 4 q^{49}+ \cdots + 20 q^{99}+O(q^{100}) \) 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
\(7\) −1.73205 + 2.00000i −0.654654 + 0.755929i
\(8\) 1.00000i 0.353553i
\(9\) 0.500000 + 0.866025i 0.166667 + 0.288675i
\(10\) 0 0
\(11\) 2.50000 − 4.33013i 0.753778 − 1.30558i −0.192201 − 0.981356i \(-0.561563\pi\)
0.945979 − 0.324227i \(-0.105104\pi\)
\(12\) 0.866025 − 0.500000i 0.250000 − 0.144338i
\(13\) − 5.00000i − 1.38675i −0.720577 − 0.693375i \(-0.756123\pi\)
0.720577 − 0.693375i \(-0.243877\pi\)
\(14\) 0.500000 − 2.59808i 0.133631 − 0.694365i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −3.46410 − 2.00000i −0.840168 − 0.485071i 0.0171533 − 0.999853i \(-0.494540\pi\)
−0.857321 + 0.514782i \(0.827873\pi\)
\(18\) −0.866025 − 0.500000i −0.204124 − 0.117851i
\(19\) −3.50000 − 6.06218i −0.802955 − 1.39076i −0.917663 − 0.397360i \(-0.869927\pi\)
0.114708 − 0.993399i \(-0.463407\pi\)
\(20\) 0 0
\(21\) −2.50000 + 0.866025i −0.545545 + 0.188982i
\(22\) 5.00000i 1.06600i
\(23\) 0.866025 − 0.500000i 0.180579 − 0.104257i −0.406986 − 0.913434i \(-0.633420\pi\)
0.587565 + 0.809177i \(0.300087\pi\)
\(24\) −0.500000 + 0.866025i −0.102062 + 0.176777i
\(25\) 0 0
\(26\) 2.50000 + 4.33013i 0.490290 + 0.849208i
\(27\) 1.00000i 0.192450i
\(28\) 0.866025 + 2.50000i 0.163663 + 0.472456i
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 1.00000 − 1.73205i 0.179605 − 0.311086i −0.762140 − 0.647412i \(-0.775851\pi\)
0.941745 + 0.336327i \(0.109185\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) 4.33013 − 2.50000i 0.753778 − 0.435194i
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −0.866025 + 0.500000i −0.142374 + 0.0821995i −0.569495 − 0.821995i \(-0.692861\pi\)
0.427121 + 0.904194i \(0.359528\pi\)
\(38\) 6.06218 + 3.50000i 0.983415 + 0.567775i
\(39\) 2.50000 − 4.33013i 0.400320 − 0.693375i
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 − 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 1.73205 − 2.00000i 0.267261 − 0.308607i
\(43\) 12.0000i 1.82998i 0.403473 + 0.914991i \(0.367803\pi\)
−0.403473 + 0.914991i \(0.632197\pi\)
\(44\) −2.50000 − 4.33013i −0.376889 − 0.652791i
\(45\) 0 0
\(46\) −0.500000 + 0.866025i −0.0737210 + 0.127688i
\(47\) 9.52628 − 5.50000i 1.38955 − 0.802257i 0.396286 − 0.918127i \(-0.370299\pi\)
0.993264 + 0.115870i \(0.0369655\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −1.00000 − 6.92820i −0.142857 − 0.989743i
\(50\) 0 0
\(51\) −2.00000 − 3.46410i −0.280056 − 0.485071i
\(52\) −4.33013 − 2.50000i −0.600481 − 0.346688i
\(53\) 7.79423 + 4.50000i 1.07062 + 0.618123i 0.928351 − 0.371706i \(-0.121227\pi\)
0.142269 + 0.989828i \(0.454560\pi\)
\(54\) −0.500000 − 0.866025i −0.0680414 − 0.117851i
\(55\) 0 0
\(56\) −2.00000 − 1.73205i −0.267261 − 0.231455i
\(57\) − 7.00000i − 0.927173i
\(58\) 0 0
\(59\) 2.00000 − 3.46410i 0.260378 − 0.450988i −0.705965 − 0.708247i \(-0.749486\pi\)
0.966342 + 0.257260i \(0.0828195\pi\)
\(60\) 0 0
\(61\) −2.00000 − 3.46410i −0.256074 − 0.443533i 0.709113 − 0.705095i \(-0.249096\pi\)
−0.965187 + 0.261562i \(0.915762\pi\)
\(62\) 2.00000i 0.254000i
\(63\) −2.59808 − 0.500000i −0.327327 − 0.0629941i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −2.50000 + 4.33013i −0.307729 + 0.533002i
\(67\) −10.3923 − 6.00000i −1.26962 − 0.733017i −0.294706 − 0.955588i \(-0.595222\pi\)
−0.974916 + 0.222571i \(0.928555\pi\)
\(68\) −3.46410 + 2.00000i −0.420084 + 0.242536i
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 − 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) −0.866025 + 0.500000i −0.102062 + 0.0589256i
\(73\) −8.66025 − 5.00000i −1.01361 − 0.585206i −0.101361 − 0.994850i \(-0.532320\pi\)
−0.912245 + 0.409644i \(0.865653\pi\)
\(74\) 0.500000 − 0.866025i 0.0581238 − 0.100673i
\(75\) 0 0
\(76\) −7.00000 −0.802955
\(77\) 4.33013 + 12.5000i 0.493464 + 1.42451i
\(78\) 5.00000i 0.566139i
\(79\) −6.00000 − 10.3923i −0.675053 − 1.16923i −0.976453 − 0.215728i \(-0.930788\pi\)
0.301401 − 0.953498i \(-0.402546\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) −4.33013 + 2.50000i −0.478183 + 0.276079i
\(83\) − 12.0000i − 1.31717i −0.752506 − 0.658586i \(-0.771155\pi\)
0.752506 − 0.658586i \(-0.228845\pi\)
\(84\) −0.500000 + 2.59808i −0.0545545 + 0.283473i
\(85\) 0 0
\(86\) −6.00000 − 10.3923i −0.646997 − 1.12063i
\(87\) 0 0
\(88\) 4.33013 + 2.50000i 0.461593 + 0.266501i
\(89\) 7.00000 + 12.1244i 0.741999 + 1.28518i 0.951584 + 0.307389i \(0.0994552\pi\)
−0.209585 + 0.977790i \(0.567211\pi\)
\(90\) 0 0
\(91\) 10.0000 + 8.66025i 1.04828 + 0.907841i
\(92\) − 1.00000i − 0.104257i
\(93\) 1.73205 − 1.00000i 0.179605 − 0.103695i
\(94\) −5.50000 + 9.52628i −0.567282 + 0.982561i
\(95\) 0 0
\(96\) 0.500000 + 0.866025i 0.0510310 + 0.0883883i
\(97\) 8.00000i 0.812277i 0.913812 + 0.406138i \(0.133125\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 4.33013 + 5.50000i 0.437409 + 0.555584i
\(99\) 5.00000 0.502519
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.g.949.1 4
5.2 odd 4 210.2.i.b.151.1 yes 2
5.3 odd 4 1050.2.i.p.151.1 2
5.4 even 2 inner 1050.2.o.g.949.2 4
7.2 even 3 inner 1050.2.o.g.499.2 4
15.2 even 4 630.2.k.g.361.1 2
20.7 even 4 1680.2.bg.d.1201.1 2
35.2 odd 12 210.2.i.b.121.1 ✓ 2
35.3 even 12 7350.2.a.a.1.1 1
35.9 even 6 inner 1050.2.o.g.499.1 4
35.12 even 12 1470.2.i.e.961.1 2
35.17 even 12 1470.2.a.o.1.1 1
35.18 odd 12 7350.2.a.u.1.1 1
35.23 odd 12 1050.2.i.p.751.1 2
35.27 even 4 1470.2.i.e.361.1 2
35.32 odd 12 1470.2.a.l.1.1 1
105.2 even 12 630.2.k.g.541.1 2
105.17 odd 12 4410.2.a.u.1.1 1
105.32 even 12 4410.2.a.j.1.1 1
140.107 even 12 1680.2.bg.d.961.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.i.b.121.1 ✓ 2 35.2 odd 12
210.2.i.b.151.1 yes 2 5.2 odd 4
630.2.k.g.361.1 2 15.2 even 4
630.2.k.g.541.1 2 105.2 even 12
1050.2.i.p.151.1 2 5.3 odd 4
1050.2.i.p.751.1 2 35.23 odd 12
1050.2.o.g.499.1 4 35.9 even 6 inner
1050.2.o.g.499.2 4 7.2 even 3 inner
1050.2.o.g.949.1 4 1.1 even 1 trivial
1050.2.o.g.949.2 4 5.4 even 2 inner
1470.2.a.l.1.1 1 35.32 odd 12
1470.2.a.o.1.1 1 35.17 even 12
1470.2.i.e.361.1 2 35.27 even 4
1470.2.i.e.961.1 2 35.12 even 12
1680.2.bg.d.961.1 2 140.107 even 12
1680.2.bg.d.1201.1 2 20.7 even 4
4410.2.a.j.1.1 1 105.32 even 12
4410.2.a.u.1.1 1 105.17 odd 12
7350.2.a.a.1.1 1 35.3 even 12
7350.2.a.u.1.1 1 35.18 odd 12