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Properties

Label 1050.2.i.p.151.1
Level $1050$
Weight $2$
Character 1050.151
Analytic conductor $8.384$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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(151,1050)] 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("1050.151"); 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, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,-1,-1,0,-2,4,-2,-1,0,5,-1,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) 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: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 151.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1050.151
Dual form 1050.2.i.p.751.1

$q$-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+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} -1.00000 q^{6} +(2.00000 + 1.73205i) q^{7} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +(2.50000 - 4.33013i) q^{11} +(-0.500000 - 0.866025i) q^{12} +5.00000 q^{13} +(-0.500000 + 2.59808i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-2.00000 + 3.46410i) q^{17} +(0.500000 - 0.866025i) q^{18} +(3.50000 + 6.06218i) q^{19} +(-2.50000 + 0.866025i) q^{21} +5.00000 q^{22} +(0.500000 + 0.866025i) q^{23} +(0.500000 - 0.866025i) q^{24} +(2.50000 + 4.33013i) q^{26} +1.00000 q^{27} +(-2.50000 + 0.866025i) q^{28} +(1.00000 - 1.73205i) q^{31} +(0.500000 - 0.866025i) q^{32} +(2.50000 + 4.33013i) q^{33} -4.00000 q^{34} +1.00000 q^{36} +(0.500000 + 0.866025i) q^{37} +(-3.50000 + 6.06218i) q^{38} +(-2.50000 + 4.33013i) q^{39} +5.00000 q^{41} +(-2.00000 - 1.73205i) q^{42} -12.0000 q^{43} +(2.50000 + 4.33013i) q^{44} +(-0.500000 + 0.866025i) q^{46} +(-5.50000 - 9.52628i) q^{47} +1.00000 q^{48} +(1.00000 + 6.92820i) q^{49} +(-2.00000 - 3.46410i) q^{51} +(-2.50000 + 4.33013i) q^{52} +(-4.50000 + 7.79423i) q^{53} +(0.500000 + 0.866025i) q^{54} +(-2.00000 - 1.73205i) q^{56} -7.00000 q^{57} +(-2.00000 + 3.46410i) q^{59} +(-2.00000 - 3.46410i) q^{61} +2.00000 q^{62} +(0.500000 - 2.59808i) q^{63} +1.00000 q^{64} +(-2.50000 + 4.33013i) q^{66} +(-6.00000 + 10.3923i) q^{67} +(-2.00000 - 3.46410i) q^{68} -1.00000 q^{69} +2.00000 q^{71} +(0.500000 + 0.866025i) q^{72} +(5.00000 - 8.66025i) q^{73} +(-0.500000 + 0.866025i) q^{74} -7.00000 q^{76} +(12.5000 - 4.33013i) q^{77} -5.00000 q^{78} +(6.00000 + 10.3923i) q^{79} +(-0.500000 + 0.866025i) q^{81} +(2.50000 + 4.33013i) q^{82} +12.0000 q^{83} +(0.500000 - 2.59808i) q^{84} +(-6.00000 - 10.3923i) q^{86} +(-2.50000 + 4.33013i) q^{88} +(-7.00000 - 12.1244i) q^{89} +(10.0000 + 8.66025i) q^{91} -1.00000 q^{92} +(1.00000 + 1.73205i) q^{93} +(5.50000 - 9.52628i) q^{94} +(0.500000 + 0.866025i) q^{96} +8.00000 q^{97} +(-5.50000 + 4.33013i) q^{98} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} - q^{4} - 2 q^{6} + 4 q^{7} - 2 q^{8} - q^{9} + 5 q^{11} - q^{12} + 10 q^{13} - q^{14} - q^{16} - 4 q^{17} + q^{18} + 7 q^{19} - 5 q^{21} + 10 q^{22} + q^{23} + q^{24} + 5 q^{26}+ \cdots - 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 + 0.866025i 0.353553 + 0.612372i
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 2.00000 + 1.73205i 0.755929 + 0.654654i
\(8\) −1.00000 −0.353553
\(9\) −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 \(-0.105104\pi\)
\(12\) −0.500000 − 0.866025i −0.144338 − 0.250000i
\(13\) 5.00000 1.38675 0.693375 − 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) −0.500000 + 2.59808i −0.133631 + 0.694365i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −2.00000 + 3.46410i −0.485071 + 0.840168i −0.999853 − 0.0171533i \(-0.994540\pi\)
0.514782 + 0.857321i \(0.327873\pi\)
\(18\) 0.500000 − 0.866025i 0.117851 − 0.204124i
\(19\) 3.50000 + 6.06218i 0.802955 + 1.39076i 0.917663 + 0.397360i \(0.130073\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) −2.50000 + 0.866025i −0.545545 + 0.188982i
\(22\) 5.00000 1.06600
\(23\) 0.500000 + 0.866025i 0.104257 + 0.180579i 0.913434 − 0.406986i \(-0.133420\pi\)
−0.809177 + 0.587565i \(0.800087\pi\)
\(24\) 0.500000 − 0.866025i 0.102062 − 0.176777i
\(25\) 0 0
\(26\) 2.50000 + 4.33013i 0.490290 + 0.849208i
\(27\) 1.00000 0.192450
\(28\) −2.50000 + 0.866025i −0.472456 + 0.163663i
\(29\) 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 \(0.109185\pi\)
\(32\) 0.500000 − 0.866025i 0.0883883 − 0.153093i
\(33\) 2.50000 + 4.33013i 0.435194 + 0.753778i
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0.500000 + 0.866025i 0.0821995 + 0.142374i 0.904194 − 0.427121i \(-0.140472\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −3.50000 + 6.06218i −0.567775 + 0.983415i
\(39\) −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 \(0.372325\pi\)
\(42\) −2.00000 − 1.73205i −0.308607 − 0.267261i
\(43\) −12.0000 −1.82998 −0.914991 − 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) 2.50000 + 4.33013i 0.376889 + 0.652791i
\(45\) 0 0
\(46\) −0.500000 + 0.866025i −0.0737210 + 0.127688i
\(47\) −5.50000 − 9.52628i −0.802257 − 1.38955i −0.918127 − 0.396286i \(-0.870299\pi\)
0.115870 − 0.993264i \(-0.463035\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 0 0
\(51\) −2.00000 − 3.46410i −0.280056 − 0.485071i
\(52\) −2.50000 + 4.33013i −0.346688 + 0.600481i
\(53\) −4.50000 + 7.79423i −0.618123 + 1.07062i 0.371706 + 0.928351i \(0.378773\pi\)
−0.989828 + 0.142269i \(0.954560\pi\)
\(54\) 0.500000 + 0.866025i 0.0680414 + 0.117851i
\(55\) 0 0
\(56\) −2.00000 − 1.73205i −0.267261 − 0.231455i
\(57\) −7.00000 −0.927173
\(58\) 0 0
\(59\) −2.00000 + 3.46410i −0.260378 + 0.450988i −0.966342 − 0.257260i \(-0.917180\pi\)
0.705965 + 0.708247i \(0.250514\pi\)
\(60\) 0 0
\(61\) −2.00000 − 3.46410i −0.256074 − 0.443533i 0.709113 − 0.705095i \(-0.249096\pi\)
−0.965187 + 0.261562i \(0.915762\pi\)
\(62\) 2.00000 0.254000
\(63\) 0.500000 − 2.59808i 0.0629941 − 0.327327i
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.50000 + 4.33013i −0.307729 + 0.533002i
\(67\) −6.00000 + 10.3923i −0.733017 + 1.26962i 0.222571 + 0.974916i \(0.428555\pi\)
−0.955588 + 0.294706i \(0.904778\pi\)
\(68\) −2.00000 − 3.46410i −0.242536 − 0.420084i
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 − 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0.500000 + 0.866025i 0.0589256 + 0.102062i
\(73\) 5.00000 − 8.66025i 0.585206 − 1.01361i −0.409644 − 0.912245i \(-0.634347\pi\)
0.994850 − 0.101361i \(-0.0323196\pi\)
\(74\) −0.500000 + 0.866025i −0.0581238 + 0.100673i
\(75\) 0 0
\(76\) −7.00000 −0.802955
\(77\) 12.5000 − 4.33013i 1.42451 − 0.493464i
\(78\) −5.00000 −0.566139
\(79\) 6.00000 + 10.3923i 0.675053 + 1.16923i 0.976453 + 0.215728i \(0.0692125\pi\)
−0.301401 + 0.953498i \(0.597454\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 2.50000 + 4.33013i 0.276079 + 0.478183i
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0.500000 − 2.59808i 0.0545545 − 0.283473i
\(85\) 0 0
\(86\) −6.00000 − 10.3923i −0.646997 − 1.12063i
\(87\) 0 0
\(88\) −2.50000 + 4.33013i −0.266501 + 0.461593i
\(89\) −7.00000 − 12.1244i −0.741999 − 1.28518i −0.951584 − 0.307389i \(-0.900545\pi\)
0.209585 − 0.977790i \(-0.432789\pi\)
\(90\) 0 0
\(91\) 10.0000 + 8.66025i 1.04828 + 0.907841i
\(92\) −1.00000 −0.104257
\(93\) 1.00000 + 1.73205i 0.103695 + 0.179605i
\(94\) 5.50000 − 9.52628i 0.567282 − 0.982561i
\(95\) 0 0
\(96\) 0.500000 + 0.866025i 0.0510310 + 0.0883883i
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −5.50000 + 4.33013i −0.555584 + 0.437409i
\(99\) −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.i.p.151.1 2
5.2 odd 4 1050.2.o.g.949.1 4
5.3 odd 4 1050.2.o.g.949.2 4
5.4 even 2 210.2.i.b.151.1 yes 2
7.2 even 3 inner 1050.2.i.p.751.1 2
7.3 odd 6 7350.2.a.a.1.1 1
7.4 even 3 7350.2.a.u.1.1 1
15.14 odd 2 630.2.k.g.361.1 2
20.19 odd 2 1680.2.bg.d.1201.1 2
35.2 odd 12 1050.2.o.g.499.2 4
35.4 even 6 1470.2.a.l.1.1 1
35.9 even 6 210.2.i.b.121.1 ✓ 2
35.19 odd 6 1470.2.i.e.961.1 2
35.23 odd 12 1050.2.o.g.499.1 4
35.24 odd 6 1470.2.a.o.1.1 1
35.34 odd 2 1470.2.i.e.361.1 2
105.44 odd 6 630.2.k.g.541.1 2
105.59 even 6 4410.2.a.u.1.1 1
105.74 odd 6 4410.2.a.j.1.1 1
140.79 odd 6 1680.2.bg.d.961.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.i.b.121.1 ✓ 2 35.9 even 6
210.2.i.b.151.1 yes 2 5.4 even 2
630.2.k.g.361.1 2 15.14 odd 2
630.2.k.g.541.1 2 105.44 odd 6
1050.2.i.p.151.1 2 1.1 even 1 trivial
1050.2.i.p.751.1 2 7.2 even 3 inner
1050.2.o.g.499.1 4 35.23 odd 12
1050.2.o.g.499.2 4 35.2 odd 12
1050.2.o.g.949.1 4 5.2 odd 4
1050.2.o.g.949.2 4 5.3 odd 4
1470.2.a.l.1.1 1 35.4 even 6
1470.2.a.o.1.1 1 35.24 odd 6
1470.2.i.e.361.1 2 35.34 odd 2
1470.2.i.e.961.1 2 35.19 odd 6
1680.2.bg.d.961.1 2 140.79 odd 6
1680.2.bg.d.1201.1 2 20.19 odd 2
4410.2.a.j.1.1 1 105.74 odd 6
4410.2.a.u.1.1 1 105.59 even 6
7350.2.a.a.1.1 1 7.3 odd 6
7350.2.a.u.1.1 1 7.4 even 3