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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [600,2,Mod(293,600)] 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("600.293"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,4,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 293.2
Root \(1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 600.293
Dual form 600.2.w.h.557.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.00000 - 1.00000i) q^{2} +(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} +2.44949 q^{6} +(3.44949 - 3.44949i) q^{7} +(-2.00000 - 2.00000i) q^{8} +3.00000i q^{9} -1.55051 q^{11} +(2.44949 - 2.44949i) q^{12} -6.89898i q^{14} -4.00000 q^{16} +(3.00000 + 3.00000i) q^{18} +8.44949 q^{21} +(-1.55051 + 1.55051i) q^{22} -4.89898i q^{24} +(-3.67423 + 3.67423i) q^{27} +(-6.89898 - 6.89898i) q^{28} +5.34847i q^{29} +4.89898 q^{31} +(-4.00000 + 4.00000i) q^{32} +(-1.89898 - 1.89898i) q^{33} +6.00000 q^{36} +(8.44949 - 8.44949i) q^{42} +3.10102i q^{44} +(-4.89898 - 4.89898i) q^{48} -16.7980i q^{49} +(2.44949 + 2.44949i) q^{53} +7.34847i q^{54} -13.7980 q^{56} +(5.34847 + 5.34847i) q^{58} +15.3485i q^{59} +(4.89898 - 4.89898i) q^{62} +(10.3485 + 10.3485i) q^{63} +8.00000i q^{64} -3.79796 q^{66} +(6.00000 - 6.00000i) q^{72} +(-11.8990 - 11.8990i) q^{73} +(-5.34847 + 5.34847i) q^{77} +14.6969i q^{79} -9.00000 q^{81} +(-4.00000 - 4.00000i) q^{83} -16.8990i q^{84} +(-6.55051 + 6.55051i) q^{87} +(3.10102 + 3.10102i) q^{88} +(6.00000 + 6.00000i) q^{93} -9.79796 q^{96} +(-8.79796 + 8.79796i) q^{97} +(-16.7980 - 16.7980i) q^{98} -4.65153i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{7} - 8 q^{8} - 16 q^{11} - 16 q^{16} + 12 q^{18} + 24 q^{21} - 16 q^{22} - 8 q^{28} - 16 q^{32} + 12 q^{33} + 24 q^{36} + 24 q^{42} - 16 q^{56} - 8 q^{58} + 12 q^{63} + 24 q^{66} + 24 q^{72}+ \cdots - 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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 − 1.00000i 0.707107 − 0.707107i
\(3\) 1.22474 + 1.22474i 0.707107 + 0.707107i
\(4\) − 2.00000i − 1.00000i
\(5\) 0 0
\(6\) 2.44949 1.00000
\(7\) 3.44949 − 3.44949i 1.30378 − 1.30378i 0.377964 − 0.925820i \(-0.376624\pi\)
0.925820 − 0.377964i \(-0.123376\pi\)
\(8\) −2.00000 − 2.00000i −0.707107 − 0.707107i
\(9\) 3.00000i 1.00000i
\(10\) 0 0
\(11\) −1.55051 −0.467496 −0.233748 − 0.972297i \(-0.575099\pi\)
−0.233748 + 0.972297i \(0.575099\pi\)
\(12\) 2.44949 − 2.44949i 0.707107 − 0.707107i
\(13\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(14\) − 6.89898i − 1.84383i
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(18\) 3.00000 + 3.00000i 0.707107 + 0.707107i
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 8.44949 1.84383
\(22\) −1.55051 + 1.55051i −0.330570 + 0.330570i
\(23\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) − 4.89898i − 1.00000i
\(25\) 0 0
\(26\) 0 0
\(27\) −3.67423 + 3.67423i −0.707107 + 0.707107i
\(28\) −6.89898 − 6.89898i −1.30378 − 1.30378i
\(29\) 5.34847i 0.993186i 0.867984 + 0.496593i \(0.165416\pi\)
−0.867984 + 0.496593i \(0.834584\pi\)
\(30\) 0 0
\(31\) 4.89898 0.879883 0.439941 − 0.898027i \(-0.354999\pi\)
0.439941 + 0.898027i \(0.354999\pi\)
\(32\) −4.00000 + 4.00000i −0.707107 + 0.707107i
\(33\) −1.89898 − 1.89898i −0.330570 − 0.330570i
\(34\) 0 0
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 8.44949 − 8.44949i 1.30378 − 1.30378i
\(43\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 3.10102i 0.467496i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) −4.89898 − 4.89898i −0.707107 − 0.707107i
\(49\) − 16.7980i − 2.39971i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.44949 + 2.44949i 0.336463 + 0.336463i 0.855034 − 0.518571i \(-0.173536\pi\)
−0.518571 + 0.855034i \(0.673536\pi\)
\(54\) 7.34847i 1.00000i
\(55\) 0 0
\(56\) −13.7980 −1.84383
\(57\) 0 0
\(58\) 5.34847 + 5.34847i 0.702288 + 0.702288i
\(59\) 15.3485i 1.99820i 0.0424110 + 0.999100i \(0.486496\pi\)
−0.0424110 + 0.999100i \(0.513504\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 4.89898 − 4.89898i 0.622171 − 0.622171i
\(63\) 10.3485 + 10.3485i 1.30378 + 1.30378i
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) −3.79796 −0.467496
\(67\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 6.00000 − 6.00000i 0.707107 − 0.707107i
\(73\) −11.8990 − 11.8990i −1.39267 − 1.39267i −0.819288 − 0.573382i \(-0.805631\pi\)
−0.573382 − 0.819288i \(-0.694369\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.34847 + 5.34847i −0.609515 + 0.609515i
\(78\) 0 0
\(79\) 14.6969i 1.65353i 0.562544 + 0.826767i \(0.309823\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) −4.00000 − 4.00000i −0.439057 − 0.439057i 0.452638 − 0.891695i \(-0.350483\pi\)
−0.891695 + 0.452638i \(0.850483\pi\)
\(84\) − 16.8990i − 1.84383i
\(85\) 0 0
\(86\) 0 0
\(87\) −6.55051 + 6.55051i −0.702288 + 0.702288i
\(88\) 3.10102 + 3.10102i 0.330570 + 0.330570i
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 6.00000 + 6.00000i 0.622171 + 0.622171i
\(94\) 0 0
\(95\) 0 0
\(96\) −9.79796 −1.00000
\(97\) −8.79796 + 8.79796i −0.893297 + 0.893297i −0.994832 − 0.101535i \(-0.967625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −16.7980 − 16.7980i −1.69685 − 1.69685i
\(99\) − 4.65153i − 0.467496i
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 600.2.w.h.293.2 4
3.2 odd 2 600.2.w.b.293.1 4
5.2 odd 4 inner 600.2.w.h.557.2 4
5.3 odd 4 120.2.w.a.77.1 yes 4
5.4 even 2 120.2.w.a.53.1 ✓ 4
8.5 even 2 600.2.w.b.293.1 4
15.2 even 4 600.2.w.b.557.1 4
15.8 even 4 120.2.w.b.77.2 yes 4
15.14 odd 2 120.2.w.b.53.2 yes 4
20.3 even 4 480.2.bi.b.17.2 4
20.19 odd 2 480.2.bi.b.113.2 4
24.5 odd 2 CM 600.2.w.h.293.2 4
40.3 even 4 480.2.bi.a.17.1 4
40.13 odd 4 120.2.w.b.77.2 yes 4
40.19 odd 2 480.2.bi.a.113.1 4
40.29 even 2 120.2.w.b.53.2 yes 4
40.37 odd 4 600.2.w.b.557.1 4
60.23 odd 4 480.2.bi.a.17.1 4
60.59 even 2 480.2.bi.a.113.1 4
120.29 odd 2 120.2.w.a.53.1 ✓ 4
120.53 even 4 120.2.w.a.77.1 yes 4
120.59 even 2 480.2.bi.b.113.2 4
120.77 even 4 inner 600.2.w.h.557.2 4
120.83 odd 4 480.2.bi.b.17.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.w.a.53.1 ✓ 4 5.4 even 2
120.2.w.a.53.1 ✓ 4 120.29 odd 2
120.2.w.a.77.1 yes 4 5.3 odd 4
120.2.w.a.77.1 yes 4 120.53 even 4
120.2.w.b.53.2 yes 4 15.14 odd 2
120.2.w.b.53.2 yes 4 40.29 even 2
120.2.w.b.77.2 yes 4 15.8 even 4
120.2.w.b.77.2 yes 4 40.13 odd 4
480.2.bi.a.17.1 4 40.3 even 4
480.2.bi.a.17.1 4 60.23 odd 4
480.2.bi.a.113.1 4 40.19 odd 2
480.2.bi.a.113.1 4 60.59 even 2
480.2.bi.b.17.2 4 20.3 even 4
480.2.bi.b.17.2 4 120.83 odd 4
480.2.bi.b.113.2 4 20.19 odd 2
480.2.bi.b.113.2 4 120.59 even 2
600.2.w.b.293.1 4 3.2 odd 2
600.2.w.b.293.1 4 8.5 even 2
600.2.w.b.557.1 4 15.2 even 4
600.2.w.b.557.1 4 40.37 odd 4
600.2.w.h.293.2 4 1.1 even 1 trivial
600.2.w.h.293.2 4 24.5 odd 2 CM
600.2.w.h.557.2 4 5.2 odd 4 inner
600.2.w.h.557.2 4 120.77 even 4 inner