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Properties

Label 1050.2.d.b.1049.4
Level $1050$
Weight $2$
Character 1050.1049
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(1049,1050)] 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("1050.1049"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1050, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-4,0] 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: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1049.4
Root \(1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1050.1049
Dual form 1050.2.d.b.1049.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +(1.22474 + 1.22474i) q^{3} +1.00000 q^{4} +(-1.22474 - 1.22474i) q^{6} +(2.44949 - 1.00000i) q^{7} -1.00000 q^{8} +3.00000i q^{9} +(1.22474 + 1.22474i) q^{12} +2.44949 q^{13} +(-2.44949 + 1.00000i) q^{14} +1.00000 q^{16} -4.89898i q^{17} -3.00000i q^{18} +2.44949i q^{19} +(4.22474 + 1.77526i) q^{21} +6.00000 q^{23} +(-1.22474 - 1.22474i) q^{24} -2.44949 q^{26} +(-3.67423 + 3.67423i) q^{27} +(2.44949 - 1.00000i) q^{28} +6.00000i q^{29} -1.00000 q^{32} +4.89898i q^{34} +3.00000i q^{36} -2.00000i q^{37} -2.44949i q^{38} +(3.00000 + 3.00000i) q^{39} +4.89898 q^{41} +(-4.22474 - 1.77526i) q^{42} -4.00000i q^{43} -6.00000 q^{46} -4.89898i q^{47} +(1.22474 + 1.22474i) q^{48} +(5.00000 - 4.89898i) q^{49} +(6.00000 - 6.00000i) q^{51} +2.44949 q^{52} +6.00000 q^{53} +(3.67423 - 3.67423i) q^{54} +(-2.44949 + 1.00000i) q^{56} +(-3.00000 + 3.00000i) q^{57} -6.00000i q^{58} -12.2474 q^{59} +12.2474i q^{61} +(3.00000 + 7.34847i) q^{63} +1.00000 q^{64} +8.00000i q^{67} -4.89898i q^{68} +(7.34847 + 7.34847i) q^{69} -3.00000i q^{72} -9.79796 q^{73} +2.00000i q^{74} +2.44949i q^{76} +(-3.00000 - 3.00000i) q^{78} +10.0000 q^{79} -9.00000 q^{81} -4.89898 q^{82} +2.44949i q^{83} +(4.22474 + 1.77526i) q^{84} +4.00000i q^{86} +(-7.34847 + 7.34847i) q^{87} +(6.00000 - 2.44949i) q^{91} +6.00000 q^{92} +4.89898i q^{94} +(-1.22474 - 1.22474i) q^{96} +4.89898 q^{97} +(-5.00000 + 4.89898i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 4 q^{16} + 12 q^{21} + 24 q^{23} - 4 q^{32} + 12 q^{39} - 12 q^{42} - 24 q^{46} + 20 q^{49} + 24 q^{51} + 24 q^{53} - 12 q^{57} + 12 q^{63} + 4 q^{64} - 12 q^{78} + 40 q^{79}+ \cdots - 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(3\) 1.22474 + 1.22474i 0.707107 + 0.707107i
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.22474 − 1.22474i −0.500000 − 0.500000i
\(7\) 2.44949 − 1.00000i 0.925820 − 0.377964i
\(8\) −1.00000 −0.353553
\(9\) 3.00000i 1.00000i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 1.22474 + 1.22474i 0.353553 + 0.353553i
\(13\) 2.44949 0.679366 0.339683 − 0.940540i \(-0.389680\pi\)
0.339683 + 0.940540i \(0.389680\pi\)
\(14\) −2.44949 + 1.00000i −0.654654 + 0.267261i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 4.89898i − 1.18818i −0.804400 − 0.594089i \(-0.797513\pi\)
0.804400 − 0.594089i \(-0.202487\pi\)
\(18\) − 3.00000i − 0.707107i
\(19\) 2.44949i 0.561951i 0.959715 + 0.280976i \(0.0906580\pi\)
−0.959715 + 0.280976i \(0.909342\pi\)
\(20\) 0 0
\(21\) 4.22474 + 1.77526i 0.921915 + 0.387392i
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −1.22474 − 1.22474i −0.250000 − 0.250000i
\(25\) 0 0
\(26\) −2.44949 −0.480384
\(27\) −3.67423 + 3.67423i −0.707107 + 0.707107i
\(28\) 2.44949 − 1.00000i 0.462910 − 0.188982i
\(29\) 6.00000i 1.11417i 0.830455 + 0.557086i \(0.188081\pi\)
−0.830455 + 0.557086i \(0.811919\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 4.89898i 0.840168i
\(35\) 0 0
\(36\) 3.00000i 0.500000i
\(37\) − 2.00000i − 0.328798i −0.986394 − 0.164399i \(-0.947432\pi\)
0.986394 − 0.164399i \(-0.0525685\pi\)
\(38\) − 2.44949i − 0.397360i
\(39\) 3.00000 + 3.00000i 0.480384 + 0.480384i
\(40\) 0 0
\(41\) 4.89898 0.765092 0.382546 − 0.923936i \(-0.375047\pi\)
0.382546 + 0.923936i \(0.375047\pi\)
\(42\) −4.22474 − 1.77526i −0.651892 − 0.273928i
\(43\) − 4.00000i − 0.609994i −0.952353 − 0.304997i \(-0.901344\pi\)
0.952353 − 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) − 4.89898i − 0.714590i −0.933992 − 0.357295i \(-0.883699\pi\)
0.933992 − 0.357295i \(-0.116301\pi\)
\(48\) 1.22474 + 1.22474i 0.176777 + 0.176777i
\(49\) 5.00000 − 4.89898i 0.714286 − 0.699854i
\(50\) 0 0
\(51\) 6.00000 − 6.00000i 0.840168 − 0.840168i
\(52\) 2.44949 0.339683
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 3.67423 − 3.67423i 0.500000 − 0.500000i
\(55\) 0 0
\(56\) −2.44949 + 1.00000i −0.327327 + 0.133631i
\(57\) −3.00000 + 3.00000i −0.397360 + 0.397360i
\(58\) − 6.00000i − 0.787839i
\(59\) −12.2474 −1.59448 −0.797241 − 0.603661i \(-0.793708\pi\)
−0.797241 + 0.603661i \(0.793708\pi\)
\(60\) 0 0
\(61\) 12.2474i 1.56813i 0.620682 + 0.784063i \(0.286856\pi\)
−0.620682 + 0.784063i \(0.713144\pi\)
\(62\) 0 0
\(63\) 3.00000 + 7.34847i 0.377964 + 0.925820i
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) − 4.89898i − 0.594089i
\(69\) 7.34847 + 7.34847i 0.884652 + 0.884652i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) − 3.00000i − 0.353553i
\(73\) −9.79796 −1.14676 −0.573382 − 0.819288i \(-0.694369\pi\)
−0.573382 + 0.819288i \(0.694369\pi\)
\(74\) 2.00000i 0.232495i
\(75\) 0 0
\(76\) 2.44949i 0.280976i
\(77\) 0 0
\(78\) −3.00000 − 3.00000i −0.339683 − 0.339683i
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) −4.89898 −0.541002
\(83\) 2.44949i 0.268866i 0.990923 + 0.134433i \(0.0429214\pi\)
−0.990923 + 0.134433i \(0.957079\pi\)
\(84\) 4.22474 + 1.77526i 0.460957 + 0.193696i
\(85\) 0 0
\(86\) 4.00000i 0.431331i
\(87\) −7.34847 + 7.34847i −0.787839 + 0.787839i
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 6.00000 − 2.44949i 0.628971 − 0.256776i
\(92\) 6.00000 0.625543
\(93\) 0 0
\(94\) 4.89898i 0.505291i
\(95\) 0 0
\(96\) −1.22474 − 1.22474i −0.125000 − 0.125000i
\(97\) 4.89898 0.497416 0.248708 − 0.968579i \(-0.419994\pi\)
0.248708 + 0.968579i \(0.419994\pi\)
\(98\) −5.00000 + 4.89898i −0.505076 + 0.494872i
\(99\) 0 0
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.d.b.1049.4 4
3.2 odd 2 1050.2.d.e.1049.3 4
5.2 odd 4 1050.2.b.b.251.2 4
5.3 odd 4 42.2.d.a.41.3 yes 4
5.4 even 2 1050.2.d.e.1049.1 4
7.6 odd 2 inner 1050.2.d.b.1049.1 4
15.2 even 4 1050.2.b.b.251.3 4
15.8 even 4 42.2.d.a.41.2 yes 4
15.14 odd 2 inner 1050.2.d.b.1049.2 4
20.3 even 4 336.2.k.b.209.3 4
21.20 even 2 1050.2.d.e.1049.2 4
35.3 even 12 294.2.f.b.215.1 8
35.13 even 4 42.2.d.a.41.4 yes 4
35.18 odd 12 294.2.f.b.215.2 8
35.23 odd 12 294.2.f.b.227.3 8
35.27 even 4 1050.2.b.b.251.1 4
35.33 even 12 294.2.f.b.227.4 8
35.34 odd 2 1050.2.d.e.1049.4 4
40.3 even 4 1344.2.k.d.1217.2 4
40.13 odd 4 1344.2.k.c.1217.3 4
45.13 odd 12 1134.2.m.g.755.1 8
45.23 even 12 1134.2.m.g.755.4 8
45.38 even 12 1134.2.m.g.377.2 8
45.43 odd 12 1134.2.m.g.377.3 8
60.23 odd 4 336.2.k.b.209.1 4
105.23 even 12 294.2.f.b.227.1 8
105.38 odd 12 294.2.f.b.215.3 8
105.53 even 12 294.2.f.b.215.4 8
105.62 odd 4 1050.2.b.b.251.4 4
105.68 odd 12 294.2.f.b.227.2 8
105.83 odd 4 42.2.d.a.41.1 ✓ 4
105.104 even 2 inner 1050.2.d.b.1049.3 4
120.53 even 4 1344.2.k.c.1217.1 4
120.83 odd 4 1344.2.k.d.1217.4 4
140.83 odd 4 336.2.k.b.209.2 4
280.13 even 4 1344.2.k.c.1217.2 4
280.83 odd 4 1344.2.k.d.1217.3 4
315.13 even 12 1134.2.m.g.755.2 8
315.83 odd 12 1134.2.m.g.377.1 8
315.223 even 12 1134.2.m.g.377.4 8
315.293 odd 12 1134.2.m.g.755.3 8
420.83 even 4 336.2.k.b.209.4 4
840.83 even 4 1344.2.k.d.1217.1 4
840.293 odd 4 1344.2.k.c.1217.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.d.a.41.1 ✓ 4 105.83 odd 4
42.2.d.a.41.2 yes 4 15.8 even 4
42.2.d.a.41.3 yes 4 5.3 odd 4
42.2.d.a.41.4 yes 4 35.13 even 4
294.2.f.b.215.1 8 35.3 even 12
294.2.f.b.215.2 8 35.18 odd 12
294.2.f.b.215.3 8 105.38 odd 12
294.2.f.b.215.4 8 105.53 even 12
294.2.f.b.227.1 8 105.23 even 12
294.2.f.b.227.2 8 105.68 odd 12
294.2.f.b.227.3 8 35.23 odd 12
294.2.f.b.227.4 8 35.33 even 12
336.2.k.b.209.1 4 60.23 odd 4
336.2.k.b.209.2 4 140.83 odd 4
336.2.k.b.209.3 4 20.3 even 4
336.2.k.b.209.4 4 420.83 even 4
1050.2.b.b.251.1 4 35.27 even 4
1050.2.b.b.251.2 4 5.2 odd 4
1050.2.b.b.251.3 4 15.2 even 4
1050.2.b.b.251.4 4 105.62 odd 4
1050.2.d.b.1049.1 4 7.6 odd 2 inner
1050.2.d.b.1049.2 4 15.14 odd 2 inner
1050.2.d.b.1049.3 4 105.104 even 2 inner
1050.2.d.b.1049.4 4 1.1 even 1 trivial
1050.2.d.e.1049.1 4 5.4 even 2
1050.2.d.e.1049.2 4 21.20 even 2
1050.2.d.e.1049.3 4 3.2 odd 2
1050.2.d.e.1049.4 4 35.34 odd 2
1134.2.m.g.377.1 8 315.83 odd 12
1134.2.m.g.377.2 8 45.38 even 12
1134.2.m.g.377.3 8 45.43 odd 12
1134.2.m.g.377.4 8 315.223 even 12
1134.2.m.g.755.1 8 45.13 odd 12
1134.2.m.g.755.2 8 315.13 even 12
1134.2.m.g.755.3 8 315.293 odd 12
1134.2.m.g.755.4 8 45.23 even 12
1344.2.k.c.1217.1 4 120.53 even 4
1344.2.k.c.1217.2 4 280.13 even 4
1344.2.k.c.1217.3 4 40.13 odd 4
1344.2.k.c.1217.4 4 840.293 odd 4
1344.2.k.d.1217.1 4 840.83 even 4
1344.2.k.d.1217.2 4 40.3 even 4
1344.2.k.d.1217.3 4 280.83 odd 4
1344.2.k.d.1217.4 4 120.83 odd 4