Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1850,2,Mod(149,1850)] 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("1850.149"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1850, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,4,0,0,-16,0,4] 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: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.3
Root \(1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1850.149
Dual form 1850.2.b.k.149.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -3.44949i q^{3} -1.00000 q^{4} +3.44949 q^{6} -2.44949i q^{7} -1.00000i q^{8} -8.89898 q^{9} -1.44949 q^{11} +3.44949i q^{12} -4.44949i q^{13} +2.44949 q^{14} +1.00000 q^{16} -1.44949i q^{17} -8.89898i q^{18} +5.00000 q^{19} -8.44949 q^{21} -1.44949i q^{22} -2.00000i q^{23} -3.44949 q^{24} +4.44949 q^{26} +20.3485i q^{27} +2.44949i q^{28} -8.89898 q^{29} -0.449490 q^{31} +1.00000i q^{32} +5.00000i q^{33} +1.44949 q^{34} +8.89898 q^{36} -1.00000i q^{37} +5.00000i q^{38} -15.3485 q^{39} +1.00000 q^{41} -8.44949i q^{42} +10.8990i q^{43} +1.44949 q^{44} +2.00000 q^{46} -9.79796i q^{47} -3.44949i q^{48} +1.00000 q^{49} -5.00000 q^{51} +4.44949i q^{52} -6.00000i q^{53} -20.3485 q^{54} -2.44949 q^{56} -17.2474i q^{57} -8.89898i q^{58} +2.00000 q^{59} -1.55051 q^{61} -0.449490i q^{62} +21.7980i q^{63} -1.00000 q^{64} -5.00000 q^{66} +9.44949i q^{67} +1.44949i q^{68} -6.89898 q^{69} -12.4495 q^{71} +8.89898i q^{72} +6.79796i q^{73} +1.00000 q^{74} -5.00000 q^{76} +3.55051i q^{77} -15.3485i q^{78} +11.7980 q^{79} +43.4949 q^{81} +1.00000i q^{82} +1.44949i q^{83} +8.44949 q^{84} -10.8990 q^{86} +30.6969i q^{87} +1.44949i q^{88} +0.348469 q^{89} -10.8990 q^{91} +2.00000i q^{92} +1.55051i q^{93} +9.79796 q^{94} +3.44949 q^{96} +14.0000i q^{97} +1.00000i q^{98} +12.8990 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{6} - 16 q^{9} + 4 q^{11} + 4 q^{16} + 20 q^{19} - 24 q^{21} - 4 q^{24} + 8 q^{26} - 16 q^{29} + 8 q^{31} - 4 q^{34} + 16 q^{36} - 32 q^{39} + 4 q^{41} - 4 q^{44} + 8 q^{46} + 4 q^{49}+ \cdots + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) − 3.44949i − 1.99156i −0.0917517 − 0.995782i \(-0.529247\pi\)
0.0917517 − 0.995782i \(-0.470753\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 3.44949 1.40825
\(7\) − 2.44949i − 0.925820i −0.886405 − 0.462910i \(-0.846805\pi\)
0.886405 − 0.462910i \(-0.153195\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −8.89898 −2.96633
\(10\) 0 0
\(11\) −1.44949 −0.437038 −0.218519 − 0.975833i \(-0.570122\pi\)
−0.218519 + 0.975833i \(0.570122\pi\)
\(12\) 3.44949i 0.995782i
\(13\) − 4.44949i − 1.23407i −0.786937 − 0.617033i \(-0.788334\pi\)
0.786937 − 0.617033i \(-0.211666\pi\)
\(14\) 2.44949 0.654654
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 1.44949i − 0.351553i −0.984430 − 0.175776i \(-0.943756\pi\)
0.984430 − 0.175776i \(-0.0562436\pi\)
\(18\) − 8.89898i − 2.09751i
\(19\) 5.00000 1.14708 0.573539 − 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) −8.44949 −1.84383
\(22\) − 1.44949i − 0.309032i
\(23\) − 2.00000i − 0.417029i −0.978019 − 0.208514i \(-0.933137\pi\)
0.978019 − 0.208514i \(-0.0668628\pi\)
\(24\) −3.44949 −0.704124
\(25\) 0 0
\(26\) 4.44949 0.872617
\(27\) 20.3485i 3.91606i
\(28\) 2.44949i 0.462910i
\(29\) −8.89898 −1.65250 −0.826250 − 0.563304i \(-0.809530\pi\)
−0.826250 + 0.563304i \(0.809530\pi\)
\(30\) 0 0
\(31\) −0.449490 −0.0807307 −0.0403654 − 0.999185i \(-0.512852\pi\)
−0.0403654 + 0.999185i \(0.512852\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 5.00000i 0.870388i
\(34\) 1.44949 0.248585
\(35\) 0 0
\(36\) 8.89898 1.48316
\(37\) − 1.00000i − 0.164399i
\(38\) 5.00000i 0.811107i
\(39\) −15.3485 −2.45772
\(40\) 0 0
\(41\) 1.00000 0.156174 0.0780869 − 0.996947i \(-0.475119\pi\)
0.0780869 + 0.996947i \(0.475119\pi\)
\(42\) − 8.44949i − 1.30378i
\(43\) 10.8990i 1.66208i 0.556214 + 0.831039i \(0.312254\pi\)
−0.556214 + 0.831039i \(0.687746\pi\)
\(44\) 1.44949 0.218519
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) − 9.79796i − 1.42918i −0.699544 − 0.714590i \(-0.746613\pi\)
0.699544 − 0.714590i \(-0.253387\pi\)
\(48\) − 3.44949i − 0.497891i
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −5.00000 −0.700140
\(52\) 4.44949i 0.617033i
\(53\) − 6.00000i − 0.824163i −0.911147 − 0.412082i \(-0.864802\pi\)
0.911147 − 0.412082i \(-0.135198\pi\)
\(54\) −20.3485 −2.76908
\(55\) 0 0
\(56\) −2.44949 −0.327327
\(57\) − 17.2474i − 2.28448i
\(58\) − 8.89898i − 1.16849i
\(59\) 2.00000 0.260378 0.130189 − 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 0 0
\(61\) −1.55051 −0.198522 −0.0992612 − 0.995061i \(-0.531648\pi\)
−0.0992612 + 0.995061i \(0.531648\pi\)
\(62\) − 0.449490i − 0.0570853i
\(63\) 21.7980i 2.74628i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −5.00000 −0.615457
\(67\) 9.44949i 1.15444i 0.816589 + 0.577219i \(0.195862\pi\)
−0.816589 + 0.577219i \(0.804138\pi\)
\(68\) 1.44949i 0.175776i
\(69\) −6.89898 −0.830540
\(70\) 0 0
\(71\) −12.4495 −1.47748 −0.738741 − 0.673989i \(-0.764579\pi\)
−0.738741 + 0.673989i \(0.764579\pi\)
\(72\) 8.89898i 1.04875i
\(73\) 6.79796i 0.795641i 0.917463 + 0.397820i \(0.130233\pi\)
−0.917463 + 0.397820i \(0.869767\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −5.00000 −0.573539
\(77\) 3.55051i 0.404618i
\(78\) − 15.3485i − 1.73787i
\(79\) 11.7980 1.32737 0.663687 − 0.748010i \(-0.268991\pi\)
0.663687 + 0.748010i \(0.268991\pi\)
\(80\) 0 0
\(81\) 43.4949 4.83277
\(82\) 1.00000i 0.110432i
\(83\) 1.44949i 0.159102i 0.996831 + 0.0795511i \(0.0253487\pi\)
−0.996831 + 0.0795511i \(0.974651\pi\)
\(84\) 8.44949 0.921915
\(85\) 0 0
\(86\) −10.8990 −1.17527
\(87\) 30.6969i 3.29106i
\(88\) 1.44949i 0.154516i
\(89\) 0.348469 0.0369377 0.0184688 − 0.999829i \(-0.494121\pi\)
0.0184688 + 0.999829i \(0.494121\pi\)
\(90\) 0 0
\(91\) −10.8990 −1.14252
\(92\) 2.00000i 0.208514i
\(93\) 1.55051i 0.160780i
\(94\) 9.79796 1.01058
\(95\) 0 0
\(96\) 3.44949 0.352062
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) 1.00000i 0.101015i
\(99\) 12.8990 1.29640
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 1850.2.b.k.149.3 4
5.2 odd 4 1850.2.a.r.1.1 ✓ 2
5.3 odd 4 1850.2.a.w.1.2 yes 2
5.4 even 2 inner 1850.2.b.k.149.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.r.1.1 ✓ 2 5.2 odd 4
1850.2.a.w.1.2 yes 2 5.3 odd 4
1850.2.b.k.149.2 4 5.4 even 2 inner
1850.2.b.k.149.3 4 1.1 even 1 trivial