Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1450,2,Mod(1,1450)] 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("1450.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1450.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,-1,2,0,-1,-5] 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: yes
Analytic conductor: \(11.5783082931\)
Analytic rank: \(1\)
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: no (minimal twist has level 290)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 1450.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.30278 q^{3} +1.00000 q^{4} +1.30278 q^{6} -4.30278 q^{7} +1.00000 q^{8} -1.30278 q^{9} -4.60555 q^{11} +1.30278 q^{12} -2.69722 q^{13} -4.30278 q^{14} +1.00000 q^{16} -6.90833 q^{17} -1.30278 q^{18} +6.60555 q^{19} -5.60555 q^{21} -4.60555 q^{22} -5.30278 q^{23} +1.30278 q^{24} -2.69722 q^{26} -5.60555 q^{27} -4.30278 q^{28} +1.00000 q^{29} +2.90833 q^{31} +1.00000 q^{32} -6.00000 q^{33} -6.90833 q^{34} -1.30278 q^{36} +11.8167 q^{37} +6.60555 q^{38} -3.51388 q^{39} -1.39445 q^{41} -5.60555 q^{42} +0.302776 q^{43} -4.60555 q^{44} -5.30278 q^{46} +1.30278 q^{48} +11.5139 q^{49} -9.00000 q^{51} -2.69722 q^{52} -6.90833 q^{53} -5.60555 q^{54} -4.30278 q^{56} +8.60555 q^{57} +1.00000 q^{58} +9.90833 q^{59} -13.9083 q^{61} +2.90833 q^{62} +5.60555 q^{63} +1.00000 q^{64} -6.00000 q^{66} -5.21110 q^{67} -6.90833 q^{68} -6.90833 q^{69} -1.30278 q^{72} +15.5139 q^{73} +11.8167 q^{74} +6.60555 q^{76} +19.8167 q^{77} -3.51388 q^{78} +5.90833 q^{79} -3.39445 q^{81} -1.39445 q^{82} +1.39445 q^{83} -5.60555 q^{84} +0.302776 q^{86} +1.30278 q^{87} -4.60555 q^{88} -7.39445 q^{89} +11.6056 q^{91} -5.30278 q^{92} +3.78890 q^{93} +1.30278 q^{96} -11.9083 q^{97} +11.5139 q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{6} - 5 q^{7} + 2 q^{8} + q^{9} - 2 q^{11} - q^{12} - 9 q^{13} - 5 q^{14} + 2 q^{16} - 3 q^{17} + q^{18} + 6 q^{19} - 4 q^{21} - 2 q^{22} - 7 q^{23} - q^{24} - 9 q^{26}+ \cdots + 12 q^{99}+O(q^{100}) \) 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 \(-0.377272\pi\)
0.376079 + 0.926588i \(0.377272\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.30278 0.531856
\(7\) −4.30278 −1.62630 −0.813148 − 0.582057i \(-0.802248\pi\)
−0.813148 + 0.582057i \(0.802248\pi\)
\(8\) 1.00000 0.353553
\(9\) −1.30278 −0.434259
\(10\) 0 0
\(11\) −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 \(-0.622027\pi\)
−0.374038 + 0.927413i \(0.622027\pi\)
\(14\) −4.30278 −1.14997
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.90833 −1.67552 −0.837758 − 0.546042i \(-0.816134\pi\)
−0.837758 + 0.546042i \(0.816134\pi\)
\(18\) −1.30278 −0.307067
\(19\) 6.60555 1.51542 0.757709 − 0.652593i \(-0.226319\pi\)
0.757709 + 0.652593i \(0.226319\pi\)
\(20\) 0 0
\(21\) −5.60555 −1.22323
\(22\) −4.60555 −0.981907
\(23\) −5.30278 −1.10571 −0.552853 − 0.833279i \(-0.686461\pi\)
−0.552853 + 0.833279i \(0.686461\pi\)
\(24\) 1.30278 0.265928
\(25\) 0 0
\(26\) −2.69722 −0.528969
\(27\) −5.60555 −1.07879
\(28\) −4.30278 −0.813148
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 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
\(35\) 0 0
\(36\) −1.30278 −0.217129
\(37\) 11.8167 1.94265 0.971323 − 0.237764i \(-0.0764145\pi\)
0.971323 + 0.237764i \(0.0764145\pi\)
\(38\) 6.60555 1.07156
\(39\) −3.51388 −0.562671
\(40\) 0 0
\(41\) −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 \(-0.492651\pi\)
0.0230864 + 0.999733i \(0.492651\pi\)
\(44\) −4.60555 −0.694313
\(45\) 0 0
\(46\) −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
\(50\) 0 0
\(51\) −9.00000 −1.26025
\(52\) −2.69722 −0.374038
\(53\) −6.90833 −0.948932 −0.474466 − 0.880274i \(-0.657359\pi\)
−0.474466 + 0.880274i \(0.657359\pi\)
\(54\) −5.60555 −0.762819
\(55\) 0 0
\(56\) −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 \(0.276867\pi\)
\(60\) 0 0
\(61\) −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
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) −5.21110 −0.636638 −0.318319 − 0.947984i \(-0.603118\pi\)
−0.318319 + 0.947984i \(0.603118\pi\)
\(68\) −6.90833 −0.837758
\(69\) −6.90833 −0.831665
\(70\) 0 0
\(71\) 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 \(-0.137699\pi\)
0.907881 + 0.419228i \(0.137699\pi\)
\(74\) 11.8167 1.37366
\(75\) 0 0
\(76\) 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 \(0.392152\pi\)
\(80\) 0 0
\(81\) −3.39445 −0.377161
\(82\) −1.39445 −0.153991
\(83\) 1.39445 0.153061 0.0765303 − 0.997067i \(-0.475616\pi\)
0.0765303 + 0.997067i \(0.475616\pi\)
\(84\) −5.60555 −0.611616
\(85\) 0 0
\(86\) 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 \(0.628184\pi\)
\(90\) 0 0
\(91\) 11.6056 1.21659
\(92\) −5.30278 −0.552853
\(93\) 3.78890 0.392890
\(94\) 0 0
\(95\) 0 0
\(96\) 1.30278 0.132964
\(97\) −11.9083 −1.20911 −0.604554 − 0.796564i \(-0.706649\pi\)
−0.604554 + 0.796564i \(0.706649\pi\)
\(98\) 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 1450.2.a.m.1.2 2
5.2 odd 4 1450.2.b.g.349.3 4
5.3 odd 4 1450.2.b.g.349.2 4
5.4 even 2 290.2.a.b.1.1 ✓ 2
15.14 odd 2 2610.2.a.v.1.2 2
20.19 odd 2 2320.2.a.i.1.2 2
40.19 odd 2 9280.2.a.bc.1.1 2
40.29 even 2 9280.2.a.z.1.2 2
145.144 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 5.4 even 2
1450.2.a.m.1.2 2 1.1 even 1 trivial
1450.2.b.g.349.2 4 5.3 odd 4
1450.2.b.g.349.3 4 5.2 odd 4
2320.2.a.i.1.2 2 20.19 odd 2
2610.2.a.v.1.2 2 15.14 odd 2
8410.2.a.r.1.2 2 145.144 even 2
9280.2.a.z.1.2 2 40.29 even 2
9280.2.a.bc.1.1 2 40.19 odd 2