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 1.1
Root \(2.30278\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.30278 q^{3} +1.00000 q^{4} -2.30278 q^{6} -0.697224 q^{7} +1.00000 q^{8} +2.30278 q^{9} +2.60555 q^{11} -2.30278 q^{12} -6.30278 q^{13} -0.697224 q^{14} +1.00000 q^{16} +3.90833 q^{17} +2.30278 q^{18} -0.605551 q^{19} +1.60555 q^{21} +2.60555 q^{22} -1.69722 q^{23} -2.30278 q^{24} -6.30278 q^{26} +1.60555 q^{27} -0.697224 q^{28} +1.00000 q^{29} -7.90833 q^{31} +1.00000 q^{32} -6.00000 q^{33} +3.90833 q^{34} +2.30278 q^{36} -9.81665 q^{37} -0.605551 q^{38} +14.5139 q^{39} -8.60555 q^{41} +1.60555 q^{42} -3.30278 q^{43} +2.60555 q^{44} -1.69722 q^{46} -2.30278 q^{48} -6.51388 q^{49} -9.00000 q^{51} -6.30278 q^{52} +3.90833 q^{53} +1.60555 q^{54} -0.697224 q^{56} +1.39445 q^{57} +1.00000 q^{58} -0.908327 q^{59} -3.09167 q^{61} -7.90833 q^{62} -1.60555 q^{63} +1.00000 q^{64} -6.00000 q^{66} +9.21110 q^{67} +3.90833 q^{68} +3.90833 q^{69} +2.30278 q^{72} -2.51388 q^{73} -9.81665 q^{74} -0.605551 q^{76} -1.81665 q^{77} +14.5139 q^{78} -4.90833 q^{79} -10.6056 q^{81} -8.60555 q^{82} +8.60555 q^{83} +1.60555 q^{84} -3.30278 q^{86} -2.30278 q^{87} +2.60555 q^{88} -14.6056 q^{89} +4.39445 q^{91} -1.69722 q^{92} +18.2111 q^{93} -2.30278 q^{96} -1.09167 q^{97} -6.51388 q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{6} - 5 q^{7} + 2 q^{8} + q^{9} - 2 q^{11} - q^{12} - 9 q^{13} - 5 q^{14} + 2 q^{16} - 3 q^{17} + q^{18} + 6 q^{19} - 4 q^{21} - 2 q^{22} - 7 q^{23} - q^{24} - 9 q^{26}+ \cdots + 12 q^{99}+O(q^{100}) \) 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
\(3\) −2.30278 −1.32951 −0.664754 − 0.747062i \(-0.731464\pi\)
−0.664754 + 0.747062i \(0.731464\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −2.30278 −0.940104
\(7\) −0.697224 −0.263526 −0.131763 − 0.991281i \(-0.542064\pi\)
−0.131763 + 0.991281i \(0.542064\pi\)
\(8\) 1.00000 0.353553
\(9\) 2.30278 0.767592
\(10\) 0 0
\(11\) 2.60555 0.785603 0.392802 − 0.919623i \(-0.371506\pi\)
0.392802 + 0.919623i \(0.371506\pi\)
\(12\) −2.30278 −0.664754
\(13\) −6.30278 −1.74808 −0.874038 − 0.485858i \(-0.838507\pi\)
−0.874038 + 0.485858i \(0.838507\pi\)
\(14\) −0.697224 −0.186341
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.90833 0.947909 0.473954 − 0.880549i \(-0.342826\pi\)
0.473954 + 0.880549i \(0.342826\pi\)
\(18\) 2.30278 0.542769
\(19\) −0.605551 −0.138923 −0.0694615 − 0.997585i \(-0.522128\pi\)
−0.0694615 + 0.997585i \(0.522128\pi\)
\(20\) 0 0
\(21\) 1.60555 0.350360
\(22\) 2.60555 0.555505
\(23\) −1.69722 −0.353896 −0.176948 − 0.984220i \(-0.556622\pi\)
−0.176948 + 0.984220i \(0.556622\pi\)
\(24\) −2.30278 −0.470052
\(25\) 0 0
\(26\) −6.30278 −1.23608
\(27\) 1.60555 0.308988
\(28\) −0.697224 −0.131763
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −7.90833 −1.42038 −0.710189 − 0.704011i \(-0.751390\pi\)
−0.710189 + 0.704011i \(0.751390\pi\)
\(32\) 1.00000 0.176777
\(33\) −6.00000 −1.04447
\(34\) 3.90833 0.670273
\(35\) 0 0
\(36\) 2.30278 0.383796
\(37\) −9.81665 −1.61385 −0.806924 − 0.590655i \(-0.798869\pi\)
−0.806924 + 0.590655i \(0.798869\pi\)
\(38\) −0.605551 −0.0982334
\(39\) 14.5139 2.32408
\(40\) 0 0
\(41\) −8.60555 −1.34396 −0.671981 − 0.740569i \(-0.734556\pi\)
−0.671981 + 0.740569i \(0.734556\pi\)
\(42\) 1.60555 0.247742
\(43\) −3.30278 −0.503669 −0.251834 − 0.967770i \(-0.581034\pi\)
−0.251834 + 0.967770i \(0.581034\pi\)
\(44\) 2.60555 0.392802
\(45\) 0 0
\(46\) −1.69722 −0.250242
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −2.30278 −0.332377
\(49\) −6.51388 −0.930554
\(50\) 0 0
\(51\) −9.00000 −1.26025
\(52\) −6.30278 −0.874038
\(53\) 3.90833 0.536850 0.268425 − 0.963301i \(-0.413497\pi\)
0.268425 + 0.963301i \(0.413497\pi\)
\(54\) 1.60555 0.218488
\(55\) 0 0
\(56\) −0.697224 −0.0931705
\(57\) 1.39445 0.184699
\(58\) 1.00000 0.131306
\(59\) −0.908327 −0.118254 −0.0591270 − 0.998250i \(-0.518832\pi\)
−0.0591270 + 0.998250i \(0.518832\pi\)
\(60\) 0 0
\(61\) −3.09167 −0.395848 −0.197924 − 0.980217i \(-0.563420\pi\)
−0.197924 + 0.980217i \(0.563420\pi\)
\(62\) −7.90833 −1.00436
\(63\) −1.60555 −0.202280
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) 9.21110 1.12532 0.562658 − 0.826690i \(-0.309779\pi\)
0.562658 + 0.826690i \(0.309779\pi\)
\(68\) 3.90833 0.473954
\(69\) 3.90833 0.470507
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 2.30278 0.271385
\(73\) −2.51388 −0.294227 −0.147114 − 0.989120i \(-0.546998\pi\)
−0.147114 + 0.989120i \(0.546998\pi\)
\(74\) −9.81665 −1.14116
\(75\) 0 0
\(76\) −0.605551 −0.0694615
\(77\) −1.81665 −0.207027
\(78\) 14.5139 1.64337
\(79\) −4.90833 −0.552230 −0.276115 − 0.961125i \(-0.589047\pi\)
−0.276115 + 0.961125i \(0.589047\pi\)
\(80\) 0 0
\(81\) −10.6056 −1.17839
\(82\) −8.60555 −0.950324
\(83\) 8.60555 0.944582 0.472291 − 0.881443i \(-0.343427\pi\)
0.472291 + 0.881443i \(0.343427\pi\)
\(84\) 1.60555 0.175180
\(85\) 0 0
\(86\) −3.30278 −0.356147
\(87\) −2.30278 −0.246883
\(88\) 2.60555 0.277753
\(89\) −14.6056 −1.54819 −0.774093 − 0.633072i \(-0.781794\pi\)
−0.774093 + 0.633072i \(0.781794\pi\)
\(90\) 0 0
\(91\) 4.39445 0.460663
\(92\) −1.69722 −0.176948
\(93\) 18.2111 1.88840
\(94\) 0 0
\(95\) 0 0
\(96\) −2.30278 −0.235026
\(97\) −1.09167 −0.110843 −0.0554213 − 0.998463i \(-0.517650\pi\)
−0.0554213 + 0.998463i \(0.517650\pi\)
\(98\) −6.51388 −0.658001
\(99\) 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.1.1 2
5.2 odd 4 1450.2.b.g.349.4 4
5.3 odd 4 1450.2.b.g.349.1 4
5.4 even 2 290.2.a.b.1.2 ✓ 2
15.14 odd 2 2610.2.a.v.1.1 2
20.19 odd 2 2320.2.a.i.1.1 2
40.19 odd 2 9280.2.a.bc.1.2 2
40.29 even 2 9280.2.a.z.1.1 2
145.144 even 2 8410.2.a.r.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
290.2.a.b.1.2 ✓ 2 5.4 even 2
1450.2.a.m.1.1 2 1.1 even 1 trivial
1450.2.b.g.349.1 4 5.3 odd 4
1450.2.b.g.349.4 4 5.2 odd 4
2320.2.a.i.1.1 2 20.19 odd 2
2610.2.a.v.1.1 2 15.14 odd 2
8410.2.a.r.1.1 2 145.144 even 2
9280.2.a.z.1.1 2 40.29 even 2
9280.2.a.bc.1.2 2 40.19 odd 2