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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1900,3,Mod(1101,1900)] 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("1900.1101"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1900.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(51.7712502285\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{6}, \sqrt{-14})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1101.3
Root \(-1.22474 - 1.87083i\) of defining polynomial
Character \(\chi\) \(=\) 1900.1101
Dual form 1900.3.e.c.1101.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.74166i q^{3} -9.79796 q^{7} -5.00000 q^{9} +4.00000 q^{11} +11.2250i q^{13} -19.5959 q^{17} +(5.00000 + 18.3303i) q^{19} -36.6606i q^{21} -9.79796 q^{23} +14.9666i q^{27} +36.6606i q^{29} -36.6606i q^{31} +14.9666i q^{33} -33.6749i q^{37} -42.0000 q^{39} +36.6606i q^{41} -68.5857 q^{43} +9.79796 q^{47} +47.0000 q^{49} -73.3212i q^{51} -56.1249i q^{53} +(-68.5857 + 18.7083i) q^{57} -73.3212i q^{59} +100.000 q^{61} +48.9898 q^{63} -11.2250i q^{67} -36.6606i q^{69} -36.6606i q^{71} -19.5959 q^{73} -39.1918 q^{77} +109.982i q^{79} -101.000 q^{81} -29.3939 q^{83} -137.171 q^{87} -146.642i q^{89} -109.982i q^{91} +137.171 q^{93} -123.475i q^{97} -20.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{9} + 16 q^{11} + 20 q^{19} - 168 q^{39} + 188 q^{49} + 400 q^{61} - 404 q^{81} - 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.74166i 1.24722i 0.781736 + 0.623610i \(0.214334\pi\)
−0.781736 + 0.623610i \(0.785666\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −9.79796 −1.39971 −0.699854 − 0.714286i \(-0.746752\pi\)
−0.699854 + 0.714286i \(0.746752\pi\)
\(8\) 0 0
\(9\) −5.00000 −0.555556
\(10\) 0 0
\(11\) 4.00000 0.363636 0.181818 − 0.983332i \(-0.441802\pi\)
0.181818 + 0.983332i \(0.441802\pi\)
\(12\) 0 0
\(13\) 11.2250i 0.863459i 0.902003 + 0.431730i \(0.142097\pi\)
−0.902003 + 0.431730i \(0.857903\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −19.5959 −1.15270 −0.576351 − 0.817203i \(-0.695524\pi\)
−0.576351 + 0.817203i \(0.695524\pi\)
\(18\) 0 0
\(19\) 5.00000 + 18.3303i 0.263158 + 0.964753i
\(20\) 0 0
\(21\) − 36.6606i − 1.74574i
\(22\) 0 0
\(23\) −9.79796 −0.425998 −0.212999 − 0.977052i \(-0.568323\pi\)
−0.212999 + 0.977052i \(0.568323\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 14.9666i 0.554320i
\(28\) 0 0
\(29\) 36.6606i 1.26416i 0.774904 + 0.632079i \(0.217798\pi\)
−0.774904 + 0.632079i \(0.782202\pi\)
\(30\) 0 0
\(31\) − 36.6606i − 1.18260i −0.806452 − 0.591300i \(-0.798615\pi\)
0.806452 − 0.591300i \(-0.201385\pi\)
\(32\) 0 0
\(33\) 14.9666i 0.453534i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 33.6749i − 0.910133i −0.890457 − 0.455066i \(-0.849616\pi\)
0.890457 − 0.455066i \(-0.150384\pi\)
\(38\) 0 0
\(39\) −42.0000 −1.07692
\(40\) 0 0
\(41\) 36.6606i 0.894161i 0.894494 + 0.447081i \(0.147536\pi\)
−0.894494 + 0.447081i \(0.852464\pi\)
\(42\) 0 0
\(43\) −68.5857 −1.59502 −0.797508 − 0.603308i \(-0.793849\pi\)
−0.797508 + 0.603308i \(0.793849\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.79796 0.208467 0.104234 − 0.994553i \(-0.466761\pi\)
0.104234 + 0.994553i \(0.466761\pi\)
\(48\) 0 0
\(49\) 47.0000 0.959184
\(50\) 0 0
\(51\) − 73.3212i − 1.43767i
\(52\) 0 0
\(53\) − 56.1249i − 1.05896i −0.848323 − 0.529480i \(-0.822387\pi\)
0.848323 − 0.529480i \(-0.177613\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −68.5857 + 18.7083i −1.20326 + 0.328216i
\(58\) 0 0
\(59\) − 73.3212i − 1.24273i −0.783520 − 0.621366i \(-0.786578\pi\)
0.783520 − 0.621366i \(-0.213422\pi\)
\(60\) 0 0
\(61\) 100.000 1.63934 0.819672 − 0.572833i \(-0.194156\pi\)
0.819672 + 0.572833i \(0.194156\pi\)
\(62\) 0 0
\(63\) 48.9898 0.777616
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 11.2250i − 0.167537i −0.996485 − 0.0837684i \(-0.973304\pi\)
0.996485 − 0.0837684i \(-0.0266956\pi\)
\(68\) 0 0
\(69\) − 36.6606i − 0.531313i
\(70\) 0 0
\(71\) − 36.6606i − 0.516347i −0.966099 − 0.258173i \(-0.916879\pi\)
0.966099 − 0.258173i \(-0.0831205\pi\)
\(72\) 0 0
\(73\) −19.5959 −0.268437 −0.134219 − 0.990952i \(-0.542852\pi\)
−0.134219 + 0.990952i \(0.542852\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −39.1918 −0.508985
\(78\) 0 0
\(79\) 109.982i 1.39217i 0.717957 + 0.696087i \(0.245077\pi\)
−0.717957 + 0.696087i \(0.754923\pi\)
\(80\) 0 0
\(81\) −101.000 −1.24691
\(82\) 0 0
\(83\) −29.3939 −0.354143 −0.177072 − 0.984198i \(-0.556662\pi\)
−0.177072 + 0.984198i \(0.556662\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −137.171 −1.57668
\(88\) 0 0
\(89\) − 146.642i − 1.64767i −0.566831 − 0.823834i \(-0.691831\pi\)
0.566831 − 0.823834i \(-0.308169\pi\)
\(90\) 0 0
\(91\) − 109.982i − 1.20859i
\(92\) 0 0
\(93\) 137.171 1.47496
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 123.475i − 1.27293i −0.771304 − 0.636467i \(-0.780395\pi\)
0.771304 − 0.636467i \(-0.219605\pi\)
\(98\) 0 0
\(99\) −20.0000 −0.202020
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 1900.3.e.c.1101.3 4
5.2 odd 4 380.3.g.a.189.3 yes 4
5.3 odd 4 380.3.g.a.189.2 yes 4
5.4 even 2 inner 1900.3.e.c.1101.2 4
15.2 even 4 3420.3.h.c.2089.2 4
15.8 even 4 3420.3.h.c.2089.3 4
19.18 odd 2 inner 1900.3.e.c.1101.1 4
95.18 even 4 380.3.g.a.189.4 yes 4
95.37 even 4 380.3.g.a.189.1 ✓ 4
95.94 odd 2 inner 1900.3.e.c.1101.4 4
285.113 odd 4 3420.3.h.c.2089.4 4
285.227 odd 4 3420.3.h.c.2089.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.g.a.189.1 ✓ 4 95.37 even 4
380.3.g.a.189.2 yes 4 5.3 odd 4
380.3.g.a.189.3 yes 4 5.2 odd 4
380.3.g.a.189.4 yes 4 95.18 even 4
1900.3.e.c.1101.1 4 19.18 odd 2 inner
1900.3.e.c.1101.2 4 5.4 even 2 inner
1900.3.e.c.1101.3 4 1.1 even 1 trivial
1900.3.e.c.1101.4 4 95.94 odd 2 inner
3420.3.h.c.2089.1 4 285.227 odd 4
3420.3.h.c.2089.2 4 15.2 even 4
3420.3.h.c.2089.3 4 15.8 even 4
3420.3.h.c.2089.4 4 285.113 odd 4