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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,2,Mod(229,380)] 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("380.229"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 229.3
Root \(0.874032i\) of defining polynomial
Character \(\chi\) \(=\) 380.229
Dual form 380.2.c.a.229.2

$q$-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)
 
\(f(q)\) \(=\) \(q+0.874032i q^{3} +2.23607 q^{5} -2.82843i q^{7} +2.23607 q^{9} -0.763932 q^{11} -5.45052i q^{13} +1.95440i q^{15} +7.40492i q^{17} +1.00000 q^{19} +2.47214 q^{21} -1.08036i q^{23} +5.00000 q^{25} +4.57649i q^{27} +4.47214 q^{29} -4.00000 q^{31} -0.667701i q^{33} -6.32456i q^{35} +2.62210i q^{37} +4.76393 q^{39} -6.00000 q^{41} +8.48528i q^{43} +5.00000 q^{45} -8.48528i q^{47} -1.00000 q^{49} -6.47214 q^{51} +2.62210i q^{53} -1.70820 q^{55} +0.874032i q^{57} +1.52786 q^{59} -11.7082 q^{61} -6.32456i q^{63} -12.1877i q^{65} -11.1074i q^{67} +0.944272 q^{69} -10.4721 q^{71} +5.24419i q^{73} +4.37016i q^{75} +2.16073i q^{77} -15.4164 q^{79} +2.70820 q^{81} +13.7295i q^{83} +16.5579i q^{85} +3.90879i q^{87} -2.94427 q^{89} -15.4164 q^{91} -3.49613i q^{93} +2.23607 q^{95} -13.9358i q^{97} -1.70820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{11} + 4 q^{19} - 8 q^{21} + 20 q^{25} - 16 q^{31} + 28 q^{39} - 24 q^{41} + 20 q^{45} - 4 q^{49} - 8 q^{51} + 20 q^{55} + 24 q^{59} - 20 q^{61} - 32 q^{69} - 24 q^{71} - 8 q^{79} - 16 q^{81}+ \cdots + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 0.874032i 0.504623i 0.967646 + 0.252311i \(0.0811907\pi\)
−0.967646 + 0.252311i \(0.918809\pi\)
\(4\) 0 0
\(5\) 2.23607 1.00000
\(6\) 0 0
\(7\) − 2.82843i − 1.06904i −0.845154 − 0.534522i \(-0.820491\pi\)
0.845154 − 0.534522i \(-0.179509\pi\)
\(8\) 0 0
\(9\) 2.23607 0.745356
\(10\) 0 0
\(11\) −0.763932 −0.230334 −0.115167 − 0.993346i \(-0.536740\pi\)
−0.115167 + 0.993346i \(0.536740\pi\)
\(12\) 0 0
\(13\) − 5.45052i − 1.51170i −0.654743 − 0.755852i \(-0.727223\pi\)
0.654743 − 0.755852i \(-0.272777\pi\)
\(14\) 0 0
\(15\) 1.95440i 0.504623i
\(16\) 0 0
\(17\) 7.40492i 1.79596i 0.440040 + 0.897978i \(0.354964\pi\)
−0.440040 + 0.897978i \(0.645036\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 2.47214 0.539464
\(22\) 0 0
\(23\) − 1.08036i − 0.225271i −0.993636 − 0.112636i \(-0.964071\pi\)
0.993636 − 0.112636i \(-0.0359293\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 4.57649i 0.880746i
\(28\) 0 0
\(29\) 4.47214 0.830455 0.415227 − 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) − 0.667701i − 0.116232i
\(34\) 0 0
\(35\) − 6.32456i − 1.06904i
\(36\) 0 0
\(37\) 2.62210i 0.431070i 0.976496 + 0.215535i \(0.0691495\pi\)
−0.976496 + 0.215535i \(0.930850\pi\)
\(38\) 0 0
\(39\) 4.76393 0.762840
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 8.48528i 1.29399i 0.762493 + 0.646997i \(0.223975\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 5.00000 0.745356
\(46\) 0 0
\(47\) − 8.48528i − 1.23771i −0.785507 − 0.618853i \(-0.787598\pi\)
0.785507 − 0.618853i \(-0.212402\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −6.47214 −0.906280
\(52\) 0 0
\(53\) 2.62210i 0.360173i 0.983651 + 0.180086i \(0.0576377\pi\)
−0.983651 + 0.180086i \(0.942362\pi\)
\(54\) 0 0
\(55\) −1.70820 −0.230334
\(56\) 0 0
\(57\) 0.874032i 0.115768i
\(58\) 0 0
\(59\) 1.52786 0.198911 0.0994555 − 0.995042i \(-0.468290\pi\)
0.0994555 + 0.995042i \(0.468290\pi\)
\(60\) 0 0
\(61\) −11.7082 −1.49908 −0.749541 − 0.661958i \(-0.769726\pi\)
−0.749541 + 0.661958i \(0.769726\pi\)
\(62\) 0 0
\(63\) − 6.32456i − 0.796819i
\(64\) 0 0
\(65\) − 12.1877i − 1.51170i
\(66\) 0 0
\(67\) − 11.1074i − 1.35698i −0.734609 − 0.678491i \(-0.762634\pi\)
0.734609 − 0.678491i \(-0.237366\pi\)
\(68\) 0 0
\(69\) 0.944272 0.113677
\(70\) 0 0
\(71\) −10.4721 −1.24281 −0.621407 − 0.783488i \(-0.713439\pi\)
−0.621407 + 0.783488i \(0.713439\pi\)
\(72\) 0 0
\(73\) 5.24419i 0.613786i 0.951744 + 0.306893i \(0.0992894\pi\)
−0.951744 + 0.306893i \(0.900711\pi\)
\(74\) 0 0
\(75\) 4.37016i 0.504623i
\(76\) 0 0
\(77\) 2.16073i 0.246238i
\(78\) 0 0
\(79\) −15.4164 −1.73448 −0.867241 − 0.497889i \(-0.834109\pi\)
−0.867241 + 0.497889i \(0.834109\pi\)
\(80\) 0 0
\(81\) 2.70820 0.300912
\(82\) 0 0
\(83\) 13.7295i 1.50701i 0.657445 + 0.753503i \(0.271637\pi\)
−0.657445 + 0.753503i \(0.728363\pi\)
\(84\) 0 0
\(85\) 16.5579i 1.79596i
\(86\) 0 0
\(87\) 3.90879i 0.419066i
\(88\) 0 0
\(89\) −2.94427 −0.312092 −0.156046 − 0.987750i \(-0.549875\pi\)
−0.156046 + 0.987750i \(0.549875\pi\)
\(90\) 0 0
\(91\) −15.4164 −1.61608
\(92\) 0 0
\(93\) − 3.49613i − 0.362532i
\(94\) 0 0
\(95\) 2.23607 0.229416
\(96\) 0 0
\(97\) − 13.9358i − 1.41497i −0.706730 − 0.707483i \(-0.749830\pi\)
0.706730 − 0.707483i \(-0.250170\pi\)
\(98\) 0 0
\(99\) −1.70820 −0.171681
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 380.2.c.a.229.3 yes 4
3.2 odd 2 3420.2.f.a.1369.1 4
4.3 odd 2 1520.2.d.f.609.2 4
5.2 odd 4 1900.2.a.j.1.3 4
5.3 odd 4 1900.2.a.j.1.2 4
5.4 even 2 inner 380.2.c.a.229.2 ✓ 4
15.14 odd 2 3420.2.f.a.1369.2 4
20.3 even 4 7600.2.a.ce.1.3 4
20.7 even 4 7600.2.a.ce.1.2 4
20.19 odd 2 1520.2.d.f.609.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.c.a.229.2 ✓ 4 5.4 even 2 inner
380.2.c.a.229.3 yes 4 1.1 even 1 trivial
1520.2.d.f.609.2 4 4.3 odd 2
1520.2.d.f.609.3 4 20.19 odd 2
1900.2.a.j.1.2 4 5.3 odd 4
1900.2.a.j.1.3 4 5.2 odd 4
3420.2.f.a.1369.1 4 3.2 odd 2
3420.2.f.a.1369.2 4 15.14 odd 2
7600.2.a.ce.1.2 4 20.7 even 4
7600.2.a.ce.1.3 4 20.3 even 4