Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [650,2,Mod(49,650)] 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("650.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(650, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,4,0] 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: \(5.19027613138\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.22581504.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 199.4
Root \(1.20036 + 0.747754i\) of defining polynomial
Character \(\chi\) \(=\) 650.199
Dual form 650.2.n.f.49.4

$q$-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} +(2.16117 + 1.24775i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(2.16117 - 1.24775i) q^{6} +(-0.165665 - 0.286941i) q^{7} -1.00000 q^{8} +(1.61378 + 2.79515i) q^{9} +(3.60108 + 2.07908i) q^{11} -2.49551i q^{12} +(3.57760 + 0.448114i) q^{13} -0.331331 q^{14} +(-0.500000 + 0.866025i) q^{16} +(-2.10108 + 1.21306i) q^{17} +3.22756 q^{18} +(-1.71006 + 0.987301i) q^{19} -0.826838i q^{21} +(3.60108 - 2.07908i) q^{22} +(0.456321 + 0.263457i) q^{23} +(-2.16117 - 1.24775i) q^{24} +(2.17688 - 2.87423i) q^{26} +0.567874i q^{27} +(-0.165665 + 0.286941i) q^{28} +(4.79066 - 8.29766i) q^{29} -4.45512i q^{31} +(0.500000 + 0.866025i) q^{32} +(5.18837 + 8.98652i) q^{33} +2.42612i q^{34} +(1.61378 - 2.79515i) q^{36} +(-2.16117 + 3.74326i) q^{37} +1.97460i q^{38} +(7.17267 + 5.43241i) q^{39} +(-9.98052 - 5.76225i) q^{41} +(-0.716063 - 0.413419i) q^{42} +(-6.97531 + 4.02720i) q^{43} -4.15817i q^{44} +(0.456321 - 0.263457i) q^{46} +2.12619 q^{47} +(-2.16117 + 1.24775i) q^{48} +(3.44511 - 5.96711i) q^{49} -6.05440 q^{51} +(-1.40072 - 3.32235i) q^{52} +10.0140i q^{53} +(0.491793 + 0.283937i) q^{54} +(0.165665 + 0.286941i) q^{56} -4.92763 q^{57} +(-4.79066 - 8.29766i) q^{58} +(2.33833 - 1.35004i) q^{59} +(4.77046 + 8.26268i) q^{61} +(-3.85824 - 2.22756i) q^{62} +(0.534695 - 0.926118i) q^{63} +1.00000 q^{64} +10.3767 q^{66} +(-5.06710 + 8.77647i) q^{67} +(2.10108 + 1.21306i) q^{68} +(0.657459 + 1.13875i) q^{69} +(-12.9670 + 7.48652i) q^{71} +(-1.61378 - 2.79515i) q^{72} -0.328354 q^{73} +(2.16117 + 3.74326i) q^{74} +(1.71006 + 0.987301i) q^{76} -1.37773i q^{77} +(8.29094 - 3.49551i) q^{78} -0.321494 q^{79} +(4.13277 - 7.15817i) q^{81} +(-9.98052 + 5.76225i) q^{82} -3.29992 q^{83} +(-0.716063 + 0.413419i) q^{84} +8.05440i q^{86} +(20.7069 - 11.9551i) q^{87} +(-3.60108 - 2.07908i) q^{88} +(-16.0240 - 9.25147i) q^{89} +(-0.464102 - 1.10080i) q^{91} -0.526914i q^{92} +(5.55889 - 9.62828i) q^{93} +(1.06310 - 1.84134i) q^{94} +2.49551i q^{96} +(-3.53618 - 6.12485i) q^{97} +(-3.44511 - 5.96711i) q^{98} +13.4207i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{4} - 8 q^{8} - 2 q^{9} + 12 q^{11} + 6 q^{13} - 4 q^{16} - 4 q^{18} - 6 q^{19} + 12 q^{22} - 24 q^{23} + 6 q^{26} + 12 q^{29} + 4 q^{32} - 2 q^{36} + 10 q^{39} - 24 q^{41} - 6 q^{42}+ \cdots - 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 − 0.866025i 0.353553 − 0.612372i
\(3\) 2.16117 + 1.24775i 1.24775 + 0.720391i 0.970661 − 0.240452i \(-0.0772957\pi\)
0.277093 + 0.960843i \(0.410629\pi\)
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) 0 0
\(6\) 2.16117 − 1.24775i 0.882295 − 0.509393i
\(7\) −0.165665 − 0.286941i −0.0626156 − 0.108453i 0.833018 − 0.553246i \(-0.186611\pi\)
−0.895634 + 0.444792i \(0.853278\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.61378 + 2.79515i 0.537926 + 0.931716i
\(10\) 0 0
\(11\) 3.60108 + 2.07908i 1.08577 + 0.626868i 0.932446 − 0.361309i \(-0.117670\pi\)
0.153320 + 0.988177i \(0.451003\pi\)
\(12\) − 2.49551i − 0.720391i
\(13\) 3.57760 + 0.448114i 0.992247 + 0.124284i
\(14\) −0.331331 −0.0885519
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −2.10108 + 1.21306i −0.509587 + 0.294210i −0.732664 − 0.680591i \(-0.761723\pi\)
0.223077 + 0.974801i \(0.428390\pi\)
\(18\) 3.22756 0.760743
\(19\) −1.71006 + 0.987301i −0.392314 + 0.226502i −0.683162 − 0.730267i \(-0.739396\pi\)
0.290849 + 0.956769i \(0.406062\pi\)
\(20\) 0 0
\(21\) − 0.826838i − 0.180431i
\(22\) 3.60108 − 2.07908i 0.767753 − 0.443262i
\(23\) 0.456321 + 0.263457i 0.0951494 + 0.0549346i 0.546820 − 0.837250i \(-0.315838\pi\)
−0.451670 + 0.892185i \(0.649172\pi\)
\(24\) −2.16117 − 1.24775i −0.441148 − 0.254697i
\(25\) 0 0
\(26\) 2.17688 − 2.87423i 0.426921 − 0.563683i
\(27\) 0.567874i 0.109287i
\(28\) −0.165665 + 0.286941i −0.0313078 + 0.0542267i
\(29\) 4.79066 − 8.29766i 0.889602 − 1.54084i 0.0492563 − 0.998786i \(-0.484315\pi\)
0.840346 − 0.542050i \(-0.182352\pi\)
\(30\) 0 0
\(31\) − 4.45512i − 0.800163i −0.916480 − 0.400081i \(-0.868982\pi\)
0.916480 − 0.400081i \(-0.131018\pi\)
\(32\) 0.500000 + 0.866025i 0.0883883 + 0.153093i
\(33\) 5.18837 + 8.98652i 0.903180 + 1.56435i
\(34\) 2.42612i 0.416076i
\(35\) 0 0
\(36\) 1.61378 − 2.79515i 0.268963 − 0.465858i
\(37\) −2.16117 + 3.74326i −0.355295 + 0.615388i −0.987168 − 0.159683i \(-0.948953\pi\)
0.631874 + 0.775071i \(0.282286\pi\)
\(38\) 1.97460i 0.320323i
\(39\) 7.17267 + 5.43241i 1.14855 + 0.869882i
\(40\) 0 0
\(41\) −9.98052 − 5.76225i −1.55869 − 0.899913i −0.997382 − 0.0723087i \(-0.976963\pi\)
−0.561312 − 0.827604i \(-0.689703\pi\)
\(42\) −0.716063 − 0.413419i −0.110491 − 0.0637920i
\(43\) −6.97531 + 4.02720i −1.06373 + 0.614142i −0.926460 − 0.376392i \(-0.877164\pi\)
−0.137265 + 0.990534i \(0.543831\pi\)
\(44\) − 4.15817i − 0.626868i
\(45\) 0 0
\(46\) 0.456321 − 0.263457i 0.0672808 − 0.0388446i
\(47\) 2.12619 0.310137 0.155069 − 0.987904i \(-0.450440\pi\)
0.155069 + 0.987904i \(0.450440\pi\)
\(48\) −2.16117 + 1.24775i −0.311938 + 0.180098i
\(49\) 3.44511 − 5.96711i 0.492159 − 0.852444i
\(50\) 0 0
\(51\) −6.05440 −0.847785
\(52\) −1.40072 − 3.32235i −0.194245 − 0.460727i
\(53\) 10.0140i 1.37553i 0.725934 + 0.687765i \(0.241408\pi\)
−0.725934 + 0.687765i \(0.758592\pi\)
\(54\) 0.491793 + 0.283937i 0.0669246 + 0.0386389i
\(55\) 0 0
\(56\) 0.165665 + 0.286941i 0.0221380 + 0.0383441i
\(57\) −4.92763 −0.652681
\(58\) −4.79066 − 8.29766i −0.629044 − 1.08954i
\(59\) 2.33833 − 1.35004i 0.304425 − 0.175760i −0.340004 − 0.940424i \(-0.610428\pi\)
0.644429 + 0.764664i \(0.277095\pi\)
\(60\) 0 0
\(61\) 4.77046 + 8.26268i 0.610795 + 1.05793i 0.991107 + 0.133069i \(0.0424833\pi\)
−0.380312 + 0.924858i \(0.624183\pi\)
\(62\) −3.85824 − 2.22756i −0.489998 − 0.282900i
\(63\) 0.534695 − 0.926118i 0.0673652 − 0.116680i
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 10.3767 1.27729
\(67\) −5.06710 + 8.77647i −0.619044 + 1.07222i 0.370616 + 0.928786i \(0.379147\pi\)
−0.989660 + 0.143430i \(0.954187\pi\)
\(68\) 2.10108 + 1.21306i 0.254793 + 0.147105i
\(69\) 0.657459 + 1.13875i 0.0791487 + 0.137090i
\(70\) 0 0
\(71\) −12.9670 + 7.48652i −1.53890 + 0.888487i −0.540001 + 0.841664i \(0.681576\pi\)
−0.998903 + 0.0468225i \(0.985090\pi\)
\(72\) −1.61378 − 2.79515i −0.190186 − 0.329411i
\(73\) −0.328354 −0.0384309 −0.0192155 − 0.999815i \(-0.506117\pi\)
−0.0192155 + 0.999815i \(0.506117\pi\)
\(74\) 2.16117 + 3.74326i 0.251231 + 0.435145i
\(75\) 0 0
\(76\) 1.71006 + 0.987301i 0.196157 + 0.113251i
\(77\) − 1.37773i − 0.157007i
\(78\) 8.29094 − 3.49551i 0.938764 − 0.395788i
\(79\) −0.321494 −0.0361709 −0.0180855 − 0.999836i \(-0.505757\pi\)
−0.0180855 + 0.999836i \(0.505757\pi\)
\(80\) 0 0
\(81\) 4.13277 − 7.15817i 0.459197 − 0.795352i
\(82\) −9.98052 + 5.76225i −1.10216 + 0.636334i
\(83\) −3.29992 −0.362214 −0.181107 − 0.983463i \(-0.557968\pi\)
−0.181107 + 0.983463i \(0.557968\pi\)
\(84\) −0.716063 + 0.413419i −0.0781289 + 0.0451077i
\(85\) 0 0
\(86\) 8.05440i 0.868528i
\(87\) 20.7069 − 11.9551i 2.22001 − 1.28172i
\(88\) −3.60108 − 2.07908i −0.383876 − 0.221631i
\(89\) −16.0240 − 9.25147i −1.69854 − 0.980654i −0.947146 − 0.320802i \(-0.896048\pi\)
−0.751396 − 0.659852i \(-0.770619\pi\)
\(90\) 0 0
\(91\) −0.464102 − 1.10080i −0.0486511 − 0.115395i
\(92\) − 0.526914i − 0.0549346i
\(93\) 5.55889 − 9.62828i 0.576430 − 0.998406i
\(94\) 1.06310 − 1.84134i 0.109650 − 0.189919i
\(95\) 0 0
\(96\) 2.49551i 0.254697i
\(97\) −3.53618 − 6.12485i −0.359045 − 0.621884i 0.628757 − 0.777602i \(-0.283564\pi\)
−0.987802 + 0.155718i \(0.950231\pi\)
\(98\) −3.44511 − 5.96711i −0.348009 − 0.602769i
\(99\) 13.4207i 1.34883i
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 650.2.n.f.199.4 8
5.2 odd 4 650.2.m.d.251.4 yes 8
5.3 odd 4 650.2.m.b.251.1 yes 8
5.4 even 2 650.2.n.c.199.1 8
13.10 even 6 650.2.n.c.49.1 8
65.7 even 12 8450.2.a.cl.1.1 4
65.23 odd 12 650.2.m.b.101.1 ✓ 8
65.32 even 12 8450.2.a.ch.1.1 4
65.33 even 12 8450.2.a.ck.1.4 4
65.49 even 6 inner 650.2.n.f.49.4 8
65.58 even 12 8450.2.a.co.1.4 4
65.62 odd 12 650.2.m.d.101.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
650.2.m.b.101.1 ✓ 8 65.23 odd 12
650.2.m.b.251.1 yes 8 5.3 odd 4
650.2.m.d.101.4 yes 8 65.62 odd 12
650.2.m.d.251.4 yes 8 5.2 odd 4
650.2.n.c.49.1 8 13.10 even 6
650.2.n.c.199.1 8 5.4 even 2
650.2.n.f.49.4 8 65.49 even 6 inner
650.2.n.f.199.4 8 1.1 even 1 trivial
8450.2.a.ch.1.1 4 65.32 even 12
8450.2.a.ck.1.4 4 65.33 even 12
8450.2.a.cl.1.1 4 65.7 even 12
8450.2.a.co.1.4 4 65.58 even 12