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Properties

Label 390.2.l.a.287.1
Level $390$
Weight $2$
Character 390.287
Analytic conductor $3.114$
Analytic rank $0$
Dimension $4$
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] = [390,2,Mod(53,390)] 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("390.53"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(390, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0] 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: \(3.11416567883\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 287.1
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 390.287
Dual form 390.2.l.a.53.1

$q$-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+(-0.707107 + 0.707107i) q^{2} +(-1.41421 - 1.00000i) q^{3} -1.00000i q^{4} +(0.707107 - 2.12132i) q^{5} +(1.70711 - 0.292893i) q^{6} +(0.585786 + 0.585786i) q^{7} +(0.707107 + 0.707107i) q^{8} +(1.00000 + 2.82843i) q^{9} +(1.00000 + 2.00000i) q^{10} -4.00000i q^{11} +(-1.00000 + 1.41421i) q^{12} +(-0.707107 + 0.707107i) q^{13} -0.828427 q^{14} +(-3.12132 + 2.29289i) q^{15} -1.00000 q^{16} +(-3.00000 + 3.00000i) q^{17} +(-2.70711 - 1.29289i) q^{18} -6.82843i q^{19} +(-2.12132 - 0.707107i) q^{20} +(-0.242641 - 1.41421i) q^{21} +(2.82843 + 2.82843i) q^{22} +(-2.82843 - 2.82843i) q^{23} +(-0.292893 - 1.70711i) q^{24} +(-4.00000 - 3.00000i) q^{25} -1.00000i q^{26} +(1.41421 - 5.00000i) q^{27} +(0.585786 - 0.585786i) q^{28} -6.00000 q^{29} +(0.585786 - 3.82843i) q^{30} +3.41421 q^{31} +(0.707107 - 0.707107i) q^{32} +(-4.00000 + 5.65685i) q^{33} -4.24264i q^{34} +(1.65685 - 0.828427i) q^{35} +(2.82843 - 1.00000i) q^{36} +(-2.58579 - 2.58579i) q^{37} +(4.82843 + 4.82843i) q^{38} +(1.70711 - 0.292893i) q^{39} +(2.00000 - 1.00000i) q^{40} -2.00000i q^{41} +(1.17157 + 0.828427i) q^{42} +(1.24264 - 1.24264i) q^{43} -4.00000 q^{44} +(6.70711 - 0.121320i) q^{45} +4.00000 q^{46} +(-2.24264 + 2.24264i) q^{47} +(1.41421 + 1.00000i) q^{48} -6.31371i q^{49} +(4.94975 - 0.707107i) q^{50} +(7.24264 - 1.24264i) q^{51} +(0.707107 + 0.707107i) q^{52} +(0.585786 + 0.585786i) q^{53} +(2.53553 + 4.53553i) q^{54} +(-8.48528 - 2.82843i) q^{55} +0.828427i q^{56} +(-6.82843 + 9.65685i) q^{57} +(4.24264 - 4.24264i) q^{58} +4.48528 q^{59} +(2.29289 + 3.12132i) q^{60} +4.82843 q^{61} +(-2.41421 + 2.41421i) q^{62} +(-1.07107 + 2.24264i) q^{63} +1.00000i q^{64} +(1.00000 + 2.00000i) q^{65} +(-1.17157 - 6.82843i) q^{66} +(-0.828427 - 0.828427i) q^{67} +(3.00000 + 3.00000i) q^{68} +(1.17157 + 6.82843i) q^{69} +(-0.585786 + 1.75736i) q^{70} +14.2426i q^{71} +(-1.29289 + 2.70711i) q^{72} +(-11.0711 + 11.0711i) q^{73} +3.65685 q^{74} +(2.65685 + 8.24264i) q^{75} -6.82843 q^{76} +(2.34315 - 2.34315i) q^{77} +(-1.00000 + 1.41421i) q^{78} -12.4853i q^{79} +(-0.707107 + 2.12132i) q^{80} +(-7.00000 + 5.65685i) q^{81} +(1.41421 + 1.41421i) q^{82} +(-6.82843 - 6.82843i) q^{83} +(-1.41421 + 0.242641i) q^{84} +(4.24264 + 8.48528i) q^{85} +1.75736i q^{86} +(8.48528 + 6.00000i) q^{87} +(2.82843 - 2.82843i) q^{88} +11.1716 q^{89} +(-4.65685 + 4.82843i) q^{90} -0.828427 q^{91} +(-2.82843 + 2.82843i) q^{92} +(-4.82843 - 3.41421i) q^{93} -3.17157i q^{94} +(-14.4853 - 4.82843i) q^{95} +(-1.70711 + 0.292893i) q^{96} +(9.07107 + 9.07107i) q^{97} +(4.46447 + 4.46447i) q^{98} +(11.3137 - 4.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{6} + 8 q^{7} + 4 q^{9} + 4 q^{10} - 4 q^{12} + 8 q^{14} - 4 q^{15} - 4 q^{16} - 12 q^{17} - 8 q^{18} + 16 q^{21} - 4 q^{24} - 16 q^{25} + 8 q^{28} - 24 q^{29} + 8 q^{30} + 8 q^{31} - 16 q^{33}+ \cdots + 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/390\mathbb{Z}\right)^\times\).

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.707107 + 0.707107i −0.500000 + 0.500000i
\(3\) −1.41421 − 1.00000i −0.816497 − 0.577350i
\(4\) − 1.00000i − 0.500000i
\(5\) 0.707107 − 2.12132i 0.316228 − 0.948683i
\(6\) 1.70711 − 0.292893i 0.696923 − 0.119573i
\(7\) 0.585786 + 0.585786i 0.221406 + 0.221406i 0.809091 − 0.587684i \(-0.199960\pi\)
−0.587684 + 0.809091i \(0.699960\pi\)
\(8\) 0.707107 + 0.707107i 0.250000 + 0.250000i
\(9\) 1.00000 + 2.82843i 0.333333 + 0.942809i
\(10\) 1.00000 + 2.00000i 0.316228 + 0.632456i
\(11\) − 4.00000i − 1.20605i −0.797724 − 0.603023i \(-0.793963\pi\)
0.797724 − 0.603023i \(-0.206037\pi\)
\(12\) −1.00000 + 1.41421i −0.288675 + 0.408248i
\(13\) −0.707107 + 0.707107i −0.196116 + 0.196116i
\(14\) −0.828427 −0.221406
\(15\) −3.12132 + 2.29289i −0.805921 + 0.592022i
\(16\) −1.00000 −0.250000
\(17\) −3.00000 + 3.00000i −0.727607 + 0.727607i −0.970143 − 0.242536i \(-0.922021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −2.70711 − 1.29289i −0.638071 − 0.304738i
\(19\) − 6.82843i − 1.56655i −0.621676 − 0.783274i \(-0.713548\pi\)
0.621676 − 0.783274i \(-0.286452\pi\)
\(20\) −2.12132 − 0.707107i −0.474342 − 0.158114i
\(21\) −0.242641 − 1.41421i −0.0529485 − 0.308607i
\(22\) 2.82843 + 2.82843i 0.603023 + 0.603023i
\(23\) −2.82843 − 2.82843i −0.589768 − 0.589768i 0.347801 − 0.937568i \(-0.386929\pi\)
−0.937568 + 0.347801i \(0.886929\pi\)
\(24\) −0.292893 − 1.70711i −0.0597866 − 0.348462i
\(25\) −4.00000 − 3.00000i −0.800000 − 0.600000i
\(26\) − 1.00000i − 0.196116i
\(27\) 1.41421 − 5.00000i 0.272166 − 0.962250i
\(28\) 0.585786 − 0.585786i 0.110703 − 0.110703i
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0.585786 − 3.82843i 0.106949 − 0.698972i
\(31\) 3.41421 0.613211 0.306605 − 0.951837i \(-0.400807\pi\)
0.306605 + 0.951837i \(0.400807\pi\)
\(32\) 0.707107 − 0.707107i 0.125000 − 0.125000i
\(33\) −4.00000 + 5.65685i −0.696311 + 0.984732i
\(34\) − 4.24264i − 0.727607i
\(35\) 1.65685 − 0.828427i 0.280059 − 0.140030i
\(36\) 2.82843 − 1.00000i 0.471405 − 0.166667i
\(37\) −2.58579 − 2.58579i −0.425101 − 0.425101i 0.461855 − 0.886956i \(-0.347184\pi\)
−0.886956 + 0.461855i \(0.847184\pi\)
\(38\) 4.82843 + 4.82843i 0.783274 + 0.783274i
\(39\) 1.70711 − 0.292893i 0.273356 − 0.0469005i
\(40\) 2.00000 − 1.00000i 0.316228 − 0.158114i
\(41\) − 2.00000i − 0.312348i −0.987730 − 0.156174i \(-0.950084\pi\)
0.987730 − 0.156174i \(-0.0499160\pi\)
\(42\) 1.17157 + 0.828427i 0.180778 + 0.127829i
\(43\) 1.24264 − 1.24264i 0.189501 − 0.189501i −0.605979 − 0.795480i \(-0.707219\pi\)
0.795480 + 0.605979i \(0.207219\pi\)
\(44\) −4.00000 −0.603023
\(45\) 6.70711 − 0.121320i 0.999836 − 0.0180854i
\(46\) 4.00000 0.589768
\(47\) −2.24264 + 2.24264i −0.327123 + 0.327123i −0.851491 − 0.524369i \(-0.824301\pi\)
0.524369 + 0.851491i \(0.324301\pi\)
\(48\) 1.41421 + 1.00000i 0.204124 + 0.144338i
\(49\) − 6.31371i − 0.901958i
\(50\) 4.94975 − 0.707107i 0.700000 − 0.100000i
\(51\) 7.24264 − 1.24264i 1.01417 − 0.174005i
\(52\) 0.707107 + 0.707107i 0.0980581 + 0.0980581i
\(53\) 0.585786 + 0.585786i 0.0804640 + 0.0804640i 0.746193 − 0.665729i \(-0.231879\pi\)
−0.665729 + 0.746193i \(0.731879\pi\)
\(54\) 2.53553 + 4.53553i 0.345042 + 0.617208i
\(55\) −8.48528 − 2.82843i −1.14416 − 0.381385i
\(56\) 0.828427i 0.110703i
\(57\) −6.82843 + 9.65685i −0.904447 + 1.27908i
\(58\) 4.24264 − 4.24264i 0.557086 − 0.557086i
\(59\) 4.48528 0.583934 0.291967 − 0.956428i \(-0.405690\pi\)
0.291967 + 0.956428i \(0.405690\pi\)
\(60\) 2.29289 + 3.12132i 0.296011 + 0.402961i
\(61\) 4.82843 0.618217 0.309108 − 0.951027i \(-0.399969\pi\)
0.309108 + 0.951027i \(0.399969\pi\)
\(62\) −2.41421 + 2.41421i −0.306605 + 0.306605i
\(63\) −1.07107 + 2.24264i −0.134942 + 0.282546i
\(64\) 1.00000i 0.125000i
\(65\) 1.00000 + 2.00000i 0.124035 + 0.248069i
\(66\) −1.17157 − 6.82843i −0.144211 − 0.840521i
\(67\) −0.828427 − 0.828427i −0.101208 − 0.101208i 0.654689 − 0.755898i \(-0.272799\pi\)
−0.755898 + 0.654689i \(0.772799\pi\)
\(68\) 3.00000 + 3.00000i 0.363803 + 0.363803i
\(69\) 1.17157 + 6.82843i 0.141041 + 0.822046i
\(70\) −0.585786 + 1.75736i −0.0700149 + 0.210045i
\(71\) 14.2426i 1.69029i 0.534537 + 0.845145i \(0.320486\pi\)
−0.534537 + 0.845145i \(0.679514\pi\)
\(72\) −1.29289 + 2.70711i −0.152369 + 0.319036i
\(73\) −11.0711 + 11.0711i −1.29577 + 1.29577i −0.364610 + 0.931160i \(0.618798\pi\)
−0.931160 + 0.364610i \(0.881202\pi\)
\(74\) 3.65685 0.425101
\(75\) 2.65685 + 8.24264i 0.306787 + 0.951778i
\(76\) −6.82843 −0.783274
\(77\) 2.34315 − 2.34315i 0.267026 − 0.267026i
\(78\) −1.00000 + 1.41421i −0.113228 + 0.160128i
\(79\) − 12.4853i − 1.40470i −0.711830 − 0.702352i \(-0.752133\pi\)
0.711830 − 0.702352i \(-0.247867\pi\)
\(80\) −0.707107 + 2.12132i −0.0790569 + 0.237171i
\(81\) −7.00000 + 5.65685i −0.777778 + 0.628539i
\(82\) 1.41421 + 1.41421i 0.156174 + 0.156174i
\(83\) −6.82843 − 6.82843i −0.749517 − 0.749517i 0.224871 − 0.974388i \(-0.427804\pi\)
−0.974388 + 0.224871i \(0.927804\pi\)
\(84\) −1.41421 + 0.242641i −0.154303 + 0.0264743i
\(85\) 4.24264 + 8.48528i 0.460179 + 0.920358i
\(86\) 1.75736i 0.189501i
\(87\) 8.48528 + 6.00000i 0.909718 + 0.643268i
\(88\) 2.82843 − 2.82843i 0.301511 − 0.301511i
\(89\) 11.1716 1.18418 0.592092 − 0.805870i \(-0.298302\pi\)
0.592092 + 0.805870i \(0.298302\pi\)
\(90\) −4.65685 + 4.82843i −0.490876 + 0.508961i
\(91\) −0.828427 −0.0868428
\(92\) −2.82843 + 2.82843i −0.294884 + 0.294884i
\(93\) −4.82843 − 3.41421i −0.500685 − 0.354037i
\(94\) − 3.17157i − 0.327123i
\(95\) −14.4853 − 4.82843i −1.48616 − 0.495386i
\(96\) −1.70711 + 0.292893i −0.174231 + 0.0298933i
\(97\) 9.07107 + 9.07107i 0.921027 + 0.921027i 0.997102 − 0.0760747i \(-0.0242388\pi\)
−0.0760747 + 0.997102i \(0.524239\pi\)
\(98\) 4.46447 + 4.46447i 0.450979 + 0.450979i
\(99\) 11.3137 − 4.00000i 1.13707 − 0.402015i
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 390.2.l.a.287.1 yes 4
3.2 odd 2 390.2.l.b.287.2 yes 4
5.3 odd 4 390.2.l.b.53.2 yes 4
15.8 even 4 inner 390.2.l.a.53.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.l.a.53.1 ✓ 4 15.8 even 4 inner
390.2.l.a.287.1 yes 4 1.1 even 1 trivial
390.2.l.b.53.2 yes 4 5.3 odd 4
390.2.l.b.287.2 yes 4 3.2 odd 2