Properties

Label 290.2.a.b.1.1
Level $290$
Weight $2$
Character 290.1
Self dual yes
Analytic conductor $2.316$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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] = [290,2,Mod(1,290)] 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("290.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(290, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 290 = 2 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 290.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,1,2,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(2.31566165862\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 290.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.30278 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.30278 q^{6} +4.30278 q^{7} -1.00000 q^{8} -1.30278 q^{9} +1.00000 q^{10} -4.60555 q^{11} -1.30278 q^{12} +2.69722 q^{13} -4.30278 q^{14} +1.30278 q^{15} +1.00000 q^{16} +6.90833 q^{17} +1.30278 q^{18} +6.60555 q^{19} -1.00000 q^{20} -5.60555 q^{21} +4.60555 q^{22} +5.30278 q^{23} +1.30278 q^{24} +1.00000 q^{25} -2.69722 q^{26} +5.60555 q^{27} +4.30278 q^{28} +1.00000 q^{29} -1.30278 q^{30} +2.90833 q^{31} -1.00000 q^{32} +6.00000 q^{33} -6.90833 q^{34} -4.30278 q^{35} -1.30278 q^{36} -11.8167 q^{37} -6.60555 q^{38} -3.51388 q^{39} +1.00000 q^{40} -1.39445 q^{41} +5.60555 q^{42} -0.302776 q^{43} -4.60555 q^{44} +1.30278 q^{45} -5.30278 q^{46} -1.30278 q^{48} +11.5139 q^{49} -1.00000 q^{50} -9.00000 q^{51} +2.69722 q^{52} +6.90833 q^{53} -5.60555 q^{54} +4.60555 q^{55} -4.30278 q^{56} -8.60555 q^{57} -1.00000 q^{58} +9.90833 q^{59} +1.30278 q^{60} -13.9083 q^{61} -2.90833 q^{62} -5.60555 q^{63} +1.00000 q^{64} -2.69722 q^{65} -6.00000 q^{66} +5.21110 q^{67} +6.90833 q^{68} -6.90833 q^{69} +4.30278 q^{70} +1.30278 q^{72} -15.5139 q^{73} +11.8167 q^{74} -1.30278 q^{75} +6.60555 q^{76} -19.8167 q^{77} +3.51388 q^{78} +5.90833 q^{79} -1.00000 q^{80} -3.39445 q^{81} +1.39445 q^{82} -1.39445 q^{83} -5.60555 q^{84} -6.90833 q^{85} +0.302776 q^{86} -1.30278 q^{87} +4.60555 q^{88} -7.39445 q^{89} -1.30278 q^{90} +11.6056 q^{91} +5.30278 q^{92} -3.78890 q^{93} -6.60555 q^{95} +1.30278 q^{96} +11.9083 q^{97} -11.5139 q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{5} - q^{6} + 5 q^{7} - 2 q^{8} + q^{9} + 2 q^{10} - 2 q^{11} + q^{12} + 9 q^{13} - 5 q^{14} - q^{15} + 2 q^{16} + 3 q^{17} - q^{18} + 6 q^{19} - 2 q^{20} - 4 q^{21}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.30278 −0.752158 −0.376079 − 0.926588i \(-0.622728\pi\)
−0.376079 + 0.926588i \(0.622728\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.30278 0.531856
\(7\) 4.30278 1.62630 0.813148 − 0.582057i \(-0.197752\pi\)
0.813148 + 0.582057i \(0.197752\pi\)
\(8\) −1.00000 −0.353553
\(9\) −1.30278 −0.434259
\(10\) 1.00000 0.316228
\(11\) −4.60555 −1.38863 −0.694313 − 0.719673i \(-0.744292\pi\)
−0.694313 + 0.719673i \(0.744292\pi\)
\(12\) −1.30278 −0.376079
\(13\) 2.69722 0.748075 0.374038 − 0.927413i \(-0.377973\pi\)
0.374038 + 0.927413i \(0.377973\pi\)
\(14\) −4.30278 −1.14997
\(15\) 1.30278 0.336375
\(16\) 1.00000 0.250000
\(17\) 6.90833 1.67552 0.837758 − 0.546042i \(-0.183866\pi\)
0.837758 + 0.546042i \(0.183866\pi\)
\(18\) 1.30278 0.307067
\(19\) 6.60555 1.51542 0.757709 − 0.652593i \(-0.226319\pi\)
0.757709 + 0.652593i \(0.226319\pi\)
\(20\) −1.00000 −0.223607
\(21\) −5.60555 −1.22323
\(22\) 4.60555 0.981907
\(23\) 5.30278 1.10571 0.552853 − 0.833279i \(-0.313539\pi\)
0.552853 + 0.833279i \(0.313539\pi\)
\(24\) 1.30278 0.265928
\(25\) 1.00000 0.200000
\(26\) −2.69722 −0.528969
\(27\) 5.60555 1.07879
\(28\) 4.30278 0.813148
\(29\) 1.00000 0.185695
\(30\) −1.30278 −0.237853
\(31\) 2.90833 0.522351 0.261175 − 0.965291i \(-0.415890\pi\)
0.261175 + 0.965291i \(0.415890\pi\)
\(32\) −1.00000 −0.176777
\(33\) 6.00000 1.04447
\(34\) −6.90833 −1.18477
\(35\) −4.30278 −0.727302
\(36\) −1.30278 −0.217129
\(37\) −11.8167 −1.94265 −0.971323 − 0.237764i \(-0.923586\pi\)
−0.971323 + 0.237764i \(0.923586\pi\)
\(38\) −6.60555 −1.07156
\(39\) −3.51388 −0.562671
\(40\) 1.00000 0.158114
\(41\) −1.39445 −0.217776 −0.108888 − 0.994054i \(-0.534729\pi\)
−0.108888 + 0.994054i \(0.534729\pi\)
\(42\) 5.60555 0.864955
\(43\) −0.302776 −0.0461729 −0.0230864 − 0.999733i \(-0.507349\pi\)
−0.0230864 + 0.999733i \(0.507349\pi\)
\(44\) −4.60555 −0.694313
\(45\) 1.30278 0.194206
\(46\) −5.30278 −0.781852
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.30278 −0.188039
\(49\) 11.5139 1.64484
\(50\) −1.00000 −0.141421
\(51\) −9.00000 −1.26025
\(52\) 2.69722 0.374038
\(53\) 6.90833 0.948932 0.474466 − 0.880274i \(-0.342641\pi\)
0.474466 + 0.880274i \(0.342641\pi\)
\(54\) −5.60555 −0.762819
\(55\) 4.60555 0.621012
\(56\) −4.30278 −0.574983
\(57\) −8.60555 −1.13983
\(58\) −1.00000 −0.131306
\(59\) 9.90833 1.28995 0.644977 − 0.764202i \(-0.276867\pi\)
0.644977 + 0.764202i \(0.276867\pi\)
\(60\) 1.30278 0.168188
\(61\) −13.9083 −1.78078 −0.890389 − 0.455200i \(-0.849568\pi\)
−0.890389 + 0.455200i \(0.849568\pi\)
\(62\) −2.90833 −0.369358
\(63\) −5.60555 −0.706233
\(64\) 1.00000 0.125000
\(65\) −2.69722 −0.334550
\(66\) −6.00000 −0.738549
\(67\) 5.21110 0.636638 0.318319 − 0.947984i \(-0.396882\pi\)
0.318319 + 0.947984i \(0.396882\pi\)
\(68\) 6.90833 0.837758
\(69\) −6.90833 −0.831665
\(70\) 4.30278 0.514280
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.30278 0.153534
\(73\) −15.5139 −1.81576 −0.907881 − 0.419228i \(-0.862301\pi\)
−0.907881 + 0.419228i \(0.862301\pi\)
\(74\) 11.8167 1.37366
\(75\) −1.30278 −0.150432
\(76\) 6.60555 0.757709
\(77\) −19.8167 −2.25832
\(78\) 3.51388 0.397868
\(79\) 5.90833 0.664739 0.332369 − 0.943149i \(-0.392152\pi\)
0.332369 + 0.943149i \(0.392152\pi\)
\(80\) −1.00000 −0.111803
\(81\) −3.39445 −0.377161
\(82\) 1.39445 0.153991
\(83\) −1.39445 −0.153061 −0.0765303 − 0.997067i \(-0.524384\pi\)
−0.0765303 + 0.997067i \(0.524384\pi\)
\(84\) −5.60555 −0.611616
\(85\) −6.90833 −0.749313
\(86\) 0.302776 0.0326491
\(87\) −1.30278 −0.139672
\(88\) 4.60555 0.490953
\(89\) −7.39445 −0.783810 −0.391905 − 0.920006i \(-0.628184\pi\)
−0.391905 + 0.920006i \(0.628184\pi\)
\(90\) −1.30278 −0.137325
\(91\) 11.6056 1.21659
\(92\) 5.30278 0.552853
\(93\) −3.78890 −0.392890
\(94\) 0 0
\(95\) −6.60555 −0.677715
\(96\) 1.30278 0.132964
\(97\) 11.9083 1.20911 0.604554 − 0.796564i \(-0.293351\pi\)
0.604554 + 0.796564i \(0.293351\pi\)
\(98\) −11.5139 −1.16308
\(99\) 6.00000 0.603023
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 290.2.a.b.1.1 ✓ 2
3.2 odd 2 2610.2.a.v.1.2 2
4.3 odd 2 2320.2.a.i.1.2 2
5.2 odd 4 1450.2.b.g.349.2 4
5.3 odd 4 1450.2.b.g.349.3 4
5.4 even 2 1450.2.a.m.1.2 2
8.3 odd 2 9280.2.a.bc.1.1 2
8.5 even 2 9280.2.a.z.1.2 2
29.28 even 2 8410.2.a.r.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
290.2.a.b.1.1 ✓ 2 1.1 even 1 trivial
1450.2.a.m.1.2 2 5.4 even 2
1450.2.b.g.349.2 4 5.2 odd 4
1450.2.b.g.349.3 4 5.3 odd 4
2320.2.a.i.1.2 2 4.3 odd 2
2610.2.a.v.1.2 2 3.2 odd 2
8410.2.a.r.1.2 2 29.28 even 2
9280.2.a.z.1.2 2 8.5 even 2
9280.2.a.bc.1.1 2 8.3 odd 2