Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1150,4,Mod(599,1150)] 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("1150.599"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-8,0,4,0,0,52,0,-64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(67.8521965066\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) 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 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.4.b.f.599.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} +1.00000i q^{3} -4.00000 q^{4} +2.00000 q^{6} +18.0000i q^{7} +8.00000i q^{8} +26.0000 q^{9} -32.0000 q^{11} -4.00000i q^{12} -47.0000i q^{13} +36.0000 q^{14} +16.0000 q^{16} -20.0000i q^{17} -52.0000i q^{18} -36.0000 q^{19} -18.0000 q^{21} +64.0000i q^{22} -23.0000i q^{23} -8.00000 q^{24} -94.0000 q^{26} +53.0000i q^{27} -72.0000i q^{28} +27.0000 q^{29} -33.0000 q^{31} -32.0000i q^{32} -32.0000i q^{33} -40.0000 q^{34} -104.000 q^{36} -56.0000i q^{37} +72.0000i q^{38} +47.0000 q^{39} -157.000 q^{41} +36.0000i q^{42} +18.0000i q^{43} +128.000 q^{44} -46.0000 q^{46} -65.0000i q^{47} +16.0000i q^{48} +19.0000 q^{49} +20.0000 q^{51} +188.000i q^{52} -14.0000i q^{53} +106.000 q^{54} -144.000 q^{56} -36.0000i q^{57} -54.0000i q^{58} +744.000 q^{59} +552.000 q^{61} +66.0000i q^{62} +468.000i q^{63} -64.0000 q^{64} -64.0000 q^{66} +156.000i q^{67} +80.0000i q^{68} +23.0000 q^{69} +699.000 q^{71} +208.000i q^{72} -609.000i q^{73} -112.000 q^{74} +144.000 q^{76} -576.000i q^{77} -94.0000i q^{78} +644.000 q^{79} +649.000 q^{81} +314.000i q^{82} +512.000i q^{83} +72.0000 q^{84} +36.0000 q^{86} +27.0000i q^{87} -256.000i q^{88} +102.000 q^{89} +846.000 q^{91} +92.0000i q^{92} -33.0000i q^{93} -130.000 q^{94} +32.0000 q^{96} -578.000i q^{97} -38.0000i q^{98} -832.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} + 4 q^{6} + 52 q^{9} - 64 q^{11} + 72 q^{14} + 32 q^{16} - 72 q^{19} - 36 q^{21} - 16 q^{24} - 188 q^{26} + 54 q^{29} - 66 q^{31} - 80 q^{34} - 208 q^{36} + 94 q^{39} - 314 q^{41} + 256 q^{44}+ \cdots - 1664 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 0.707107i
\(3\) 1.00000i 0.192450i 0.995360 + 0.0962250i \(0.0306768\pi\)
−0.995360 + 0.0962250i \(0.969323\pi\)
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) 2.00000 0.136083
\(7\) 18.0000i 0.971909i 0.873984 + 0.485954i \(0.161528\pi\)
−0.873984 + 0.485954i \(0.838472\pi\)
\(8\) 8.00000i 0.353553i
\(9\) 26.0000 0.962963
\(10\) 0 0
\(11\) −32.0000 −0.877124 −0.438562 − 0.898701i \(-0.644512\pi\)
−0.438562 + 0.898701i \(0.644512\pi\)
\(12\) − 4.00000i − 0.0962250i
\(13\) − 47.0000i − 1.00273i −0.865237 − 0.501364i \(-0.832832\pi\)
0.865237 − 0.501364i \(-0.167168\pi\)
\(14\) 36.0000 0.687243
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) − 20.0000i − 0.285336i −0.989771 − 0.142668i \(-0.954432\pi\)
0.989771 − 0.142668i \(-0.0455681\pi\)
\(18\) − 52.0000i − 0.680918i
\(19\) −36.0000 −0.434682 −0.217341 − 0.976096i \(-0.569738\pi\)
−0.217341 + 0.976096i \(0.569738\pi\)
\(20\) 0 0
\(21\) −18.0000 −0.187044
\(22\) 64.0000i 0.620220i
\(23\) − 23.0000i − 0.208514i
\(24\) −8.00000 −0.0680414
\(25\) 0 0
\(26\) −94.0000 −0.709035
\(27\) 53.0000i 0.377772i
\(28\) − 72.0000i − 0.485954i
\(29\) 27.0000 0.172889 0.0864444 − 0.996257i \(-0.472450\pi\)
0.0864444 + 0.996257i \(0.472450\pi\)
\(30\) 0 0
\(31\) −33.0000 −0.191193 −0.0955964 − 0.995420i \(-0.530476\pi\)
−0.0955964 + 0.995420i \(0.530476\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) − 32.0000i − 0.168803i
\(34\) −40.0000 −0.201763
\(35\) 0 0
\(36\) −104.000 −0.481481
\(37\) − 56.0000i − 0.248820i −0.992231 − 0.124410i \(-0.960296\pi\)
0.992231 − 0.124410i \(-0.0397038\pi\)
\(38\) 72.0000i 0.307367i
\(39\) 47.0000 0.192975
\(40\) 0 0
\(41\) −157.000 −0.598031 −0.299016 − 0.954248i \(-0.596658\pi\)
−0.299016 + 0.954248i \(0.596658\pi\)
\(42\) 36.0000i 0.132260i
\(43\) 18.0000i 0.0638366i 0.999490 + 0.0319183i \(0.0101616\pi\)
−0.999490 + 0.0319183i \(0.989838\pi\)
\(44\) 128.000 0.438562
\(45\) 0 0
\(46\) −46.0000 −0.147442
\(47\) − 65.0000i − 0.201728i −0.994900 − 0.100864i \(-0.967839\pi\)
0.994900 − 0.100864i \(-0.0321607\pi\)
\(48\) 16.0000i 0.0481125i
\(49\) 19.0000 0.0553936
\(50\) 0 0
\(51\) 20.0000 0.0549129
\(52\) 188.000i 0.501364i
\(53\) − 14.0000i − 0.0362839i −0.999835 − 0.0181420i \(-0.994225\pi\)
0.999835 − 0.0181420i \(-0.00577508\pi\)
\(54\) 106.000 0.267125
\(55\) 0 0
\(56\) −144.000 −0.343622
\(57\) − 36.0000i − 0.0836547i
\(58\) − 54.0000i − 0.122251i
\(59\) 744.000 1.64170 0.820852 − 0.571141i \(-0.193499\pi\)
0.820852 + 0.571141i \(0.193499\pi\)
\(60\) 0 0
\(61\) 552.000 1.15863 0.579314 − 0.815104i \(-0.303320\pi\)
0.579314 + 0.815104i \(0.303320\pi\)
\(62\) 66.0000i 0.135194i
\(63\) 468.000i 0.935912i
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) −64.0000 −0.119361
\(67\) 156.000i 0.284454i 0.989834 + 0.142227i \(0.0454263\pi\)
−0.989834 + 0.142227i \(0.954574\pi\)
\(68\) 80.0000i 0.142668i
\(69\) 23.0000 0.0401286
\(70\) 0 0
\(71\) 699.000 1.16839 0.584197 − 0.811612i \(-0.301409\pi\)
0.584197 + 0.811612i \(0.301409\pi\)
\(72\) 208.000i 0.340459i
\(73\) − 609.000i − 0.976412i −0.872728 − 0.488206i \(-0.837652\pi\)
0.872728 − 0.488206i \(-0.162348\pi\)
\(74\) −112.000 −0.175942
\(75\) 0 0
\(76\) 144.000 0.217341
\(77\) − 576.000i − 0.852484i
\(78\) − 94.0000i − 0.136454i
\(79\) 644.000 0.917160 0.458580 − 0.888653i \(-0.348358\pi\)
0.458580 + 0.888653i \(0.348358\pi\)
\(80\) 0 0
\(81\) 649.000 0.890261
\(82\) 314.000i 0.422872i
\(83\) 512.000i 0.677100i 0.940948 + 0.338550i \(0.109936\pi\)
−0.940948 + 0.338550i \(0.890064\pi\)
\(84\) 72.0000 0.0935220
\(85\) 0 0
\(86\) 36.0000 0.0451393
\(87\) 27.0000i 0.0332725i
\(88\) − 256.000i − 0.310110i
\(89\) 102.000 0.121483 0.0607415 − 0.998154i \(-0.480653\pi\)
0.0607415 + 0.998154i \(0.480653\pi\)
\(90\) 0 0
\(91\) 846.000 0.974559
\(92\) 92.0000i 0.104257i
\(93\) − 33.0000i − 0.0367951i
\(94\) −130.000 −0.142643
\(95\) 0 0
\(96\) 32.0000 0.0340207
\(97\) − 578.000i − 0.605021i −0.953146 − 0.302510i \(-0.902175\pi\)
0.953146 − 0.302510i \(-0.0978247\pi\)
\(98\) − 38.0000i − 0.0391692i
\(99\) −832.000 −0.844638
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 1150.4.b.f.599.1 2
5.2 odd 4 230.4.a.e.1.1 ✓ 1
5.3 odd 4 1150.4.a.b.1.1 1
5.4 even 2 inner 1150.4.b.f.599.2 2
15.2 even 4 2070.4.a.e.1.1 1
20.7 even 4 1840.4.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.e.1.1 ✓ 1 5.2 odd 4
1150.4.a.b.1.1 1 5.3 odd 4
1150.4.b.f.599.1 2 1.1 even 1 trivial
1150.4.b.f.599.2 2 5.4 even 2 inner
1840.4.a.d.1.1 1 20.7 even 4
2070.4.a.e.1.1 1 15.2 even 4