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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,2,Mod(13,380)] 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("380.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([0, 27, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.bh (of order \(36\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(10\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 33.5
Character \(\chi\) \(=\) 380.33
Dual form 380.2.bh.a.357.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.0411373 + 0.0587502i) q^{3} +(-2.20598 + 0.365568i) q^{5} +(0.553309 + 0.148259i) q^{7} +(1.02430 - 2.81424i) q^{9} +(2.49421 - 4.32010i) q^{11} +(4.89698 + 3.42890i) q^{13} +(-0.112225 - 0.114563i) q^{15} +(1.10088 - 2.36084i) q^{17} +(3.41080 + 2.71412i) q^{19} +(0.0140514 + 0.0386060i) q^{21} +(4.15825 + 0.363800i) q^{23} +(4.73272 - 1.61287i) q^{25} +(0.415305 - 0.111281i) q^{27} +(-5.28906 - 1.92506i) q^{29} +(-5.80028 + 3.34880i) q^{31} +(0.356412 - 0.0311820i) q^{33} +(-1.27479 - 0.124784i) q^{35} +(-5.89970 - 5.89970i) q^{37} +0.428755i q^{39} +(-1.05503 - 0.186031i) q^{41} +(0.00412207 + 0.0471155i) q^{43} +(-1.23079 + 6.58263i) q^{45} +(10.8196 - 5.04526i) q^{47} +(-5.77801 - 3.33593i) q^{49} +(0.183987 - 0.0324418i) q^{51} +(-0.268926 + 3.07384i) q^{53} +(-3.92290 + 10.4419i) q^{55} +(-0.0191439 + 0.312037i) q^{57} +(6.31979 - 2.30021i) q^{59} +(-6.37870 + 5.35237i) q^{61} +(0.983991 - 1.40528i) q^{63} +(-12.0562 - 5.77392i) q^{65} +(5.22905 + 11.2137i) q^{67} +(0.149686 + 0.259264i) q^{69} +(-0.975910 + 1.16304i) q^{71} +(-2.22190 + 1.55579i) q^{73} +(0.289448 + 0.211699i) q^{75} +(2.02056 - 2.02056i) q^{77} +(1.81235 - 10.2784i) q^{79} +(-6.85896 - 5.75535i) q^{81} +(-1.33629 + 4.98709i) q^{83} +(-1.56547 + 5.61041i) q^{85} +(-0.104480 - 0.389925i) q^{87} +(-1.70407 - 9.66428i) q^{89} +(2.20118 + 2.62326i) q^{91} +(-0.435351 - 0.203007i) q^{93} +(-8.51636 - 4.74042i) q^{95} +(-10.0585 - 4.69035i) q^{97} +(-9.60300 - 11.4444i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 6 q^{7} + 18 q^{15} - 18 q^{17} + 48 q^{21} - 36 q^{23} - 24 q^{25} - 60 q^{33} - 18 q^{35} - 12 q^{41} - 36 q^{43} + 18 q^{45} - 24 q^{47} + 96 q^{51} - 18 q^{53} + 72 q^{55} - 6 q^{57} - 24 q^{61}+ \cdots - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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
\(3\) 0.0411373 + 0.0587502i 0.0237507 + 0.0339194i 0.830852 − 0.556494i \(-0.187854\pi\)
−0.807101 + 0.590413i \(0.798965\pi\)
\(4\) 0 0
\(5\) −2.20598 + 0.365568i −0.986546 + 0.163487i
\(6\) 0 0
\(7\) 0.553309 + 0.148259i 0.209131 + 0.0560365i 0.361864 − 0.932231i \(-0.382141\pi\)
−0.152733 + 0.988268i \(0.548807\pi\)
\(8\) 0 0
\(9\) 1.02430 − 2.81424i 0.341434 − 0.938081i
\(10\) 0 0
\(11\) 2.49421 − 4.32010i 0.752034 − 1.30256i −0.194802 − 0.980843i \(-0.562406\pi\)
0.946836 − 0.321718i \(-0.104260\pi\)
\(12\) 0 0
\(13\) 4.89698 + 3.42890i 1.35818 + 0.951007i 0.999848 + 0.0174117i \(0.00554259\pi\)
0.358330 + 0.933595i \(0.383346\pi\)
\(14\) 0 0
\(15\) −0.112225 − 0.114563i −0.0289765 − 0.0295802i
\(16\) 0 0
\(17\) 1.10088 − 2.36084i 0.267002 − 0.572587i −0.726400 − 0.687272i \(-0.758808\pi\)
0.993402 + 0.114685i \(0.0365858\pi\)
\(18\) 0 0
\(19\) 3.41080 + 2.71412i 0.782491 + 0.622662i
\(20\) 0 0
\(21\) 0.0140514 + 0.0386060i 0.00306627 + 0.00842451i
\(22\) 0 0
\(23\) 4.15825 + 0.363800i 0.867055 + 0.0758575i 0.511997 − 0.858987i \(-0.328906\pi\)
0.355057 + 0.934844i \(0.384461\pi\)
\(24\) 0 0
\(25\) 4.73272 − 1.61287i 0.946544 − 0.322574i
\(26\) 0 0
\(27\) 0.415305 − 0.111281i 0.0799255 − 0.0214160i
\(28\) 0 0
\(29\) −5.28906 − 1.92506i −0.982153 − 0.357474i −0.199476 − 0.979903i \(-0.563924\pi\)
−0.782677 + 0.622428i \(0.786146\pi\)
\(30\) 0 0
\(31\) −5.80028 + 3.34880i −1.04176 + 0.601461i −0.920332 − 0.391139i \(-0.872081\pi\)
−0.121430 + 0.992600i \(0.538748\pi\)
\(32\) 0 0
\(33\) 0.356412 − 0.0311820i 0.0620434 − 0.00542810i
\(34\) 0 0
\(35\) −1.27479 − 0.124784i −0.215478 − 0.0210924i
\(36\) 0 0
\(37\) −5.89970 − 5.89970i −0.969905 − 0.969905i 0.0296550 − 0.999560i \(-0.490559\pi\)
−0.999560 + 0.0296550i \(0.990559\pi\)
\(38\) 0 0
\(39\) 0.428755i 0.0686557i
\(40\) 0 0
\(41\) −1.05503 − 0.186031i −0.164768 − 0.0290531i 0.0906553 − 0.995882i \(-0.471104\pi\)
−0.255424 + 0.966829i \(0.582215\pi\)
\(42\) 0 0
\(43\) 0.00412207 + 0.0471155i 0.000628610 + 0.00718504i 0.996503 − 0.0835627i \(-0.0266299\pi\)
−0.995874 + 0.0907477i \(0.971074\pi\)
\(44\) 0 0
\(45\) −1.23079 + 6.58263i −0.183476 + 0.981280i
\(46\) 0 0
\(47\) 10.8196 − 5.04526i 1.57820 − 0.735927i 0.581232 − 0.813738i \(-0.302571\pi\)
0.996968 + 0.0778110i \(0.0247931\pi\)
\(48\) 0 0
\(49\) −5.77801 − 3.33593i −0.825430 − 0.476562i
\(50\) 0 0
\(51\) 0.183987 − 0.0324418i 0.0257633 − 0.00454277i
\(52\) 0 0
\(53\) −0.268926 + 3.07384i −0.0369398 + 0.422224i 0.955099 + 0.296288i \(0.0957489\pi\)
−0.992038 + 0.125936i \(0.959807\pi\)
\(54\) 0 0
\(55\) −3.92290 + 10.4419i −0.528964 + 1.40798i
\(56\) 0 0
\(57\) −0.0191439 + 0.312037i −0.00253567 + 0.0413303i
\(58\) 0 0
\(59\) 6.31979 − 2.30021i 0.822766 − 0.299462i 0.103880 − 0.994590i \(-0.466874\pi\)
0.718886 + 0.695128i \(0.244652\pi\)
\(60\) 0 0
\(61\) −6.37870 + 5.35237i −0.816709 + 0.685300i −0.952199 − 0.305478i \(-0.901184\pi\)
0.135490 + 0.990779i \(0.456739\pi\)
\(62\) 0 0
\(63\) 0.983991 − 1.40528i 0.123971 − 0.177049i
\(64\) 0 0
\(65\) −12.0562 − 5.77392i −1.49538 − 0.716167i
\(66\) 0 0
\(67\) 5.22905 + 11.2137i 0.638830 + 1.36997i 0.912868 + 0.408256i \(0.133863\pi\)
−0.274038 + 0.961719i \(0.588359\pi\)
\(68\) 0 0
\(69\) 0.149686 + 0.259264i 0.0180201 + 0.0312117i
\(70\) 0 0
\(71\) −0.975910 + 1.16304i −0.115819 + 0.138028i −0.820839 − 0.571160i \(-0.806494\pi\)
0.705020 + 0.709188i \(0.250938\pi\)
\(72\) 0 0
\(73\) −2.22190 + 1.55579i −0.260053 + 0.182091i −0.696331 − 0.717721i \(-0.745185\pi\)
0.436278 + 0.899812i \(0.356297\pi\)
\(74\) 0 0
\(75\) 0.289448 + 0.211699i 0.0334226 + 0.0244449i
\(76\) 0 0
\(77\) 2.02056 − 2.02056i 0.230264 − 0.230264i
\(78\) 0 0
\(79\) 1.81235 − 10.2784i 0.203905 − 1.15641i −0.695247 − 0.718771i \(-0.744705\pi\)
0.899153 − 0.437635i \(-0.144184\pi\)
\(80\) 0 0
\(81\) −6.85896 − 5.75535i −0.762106 − 0.639483i
\(82\) 0 0
\(83\) −1.33629 + 4.98709i −0.146676 + 0.547404i 0.852999 + 0.521913i \(0.174782\pi\)
−0.999675 + 0.0254909i \(0.991885\pi\)
\(84\) 0 0
\(85\) −1.56547 + 5.61041i −0.169799 + 0.608535i
\(86\) 0 0
\(87\) −0.104480 − 0.389925i −0.0112014 − 0.0418043i
\(88\) 0 0
\(89\) −1.70407 − 9.66428i −0.180631 − 1.02441i −0.931441 − 0.363892i \(-0.881448\pi\)
0.750810 − 0.660519i \(-0.229664\pi\)
\(90\) 0 0
\(91\) 2.20118 + 2.62326i 0.230746 + 0.274993i
\(92\) 0 0
\(93\) −0.435351 − 0.203007i −0.0451438 − 0.0210509i
\(94\) 0 0
\(95\) −8.51636 − 4.74042i −0.873760 − 0.486357i
\(96\) 0 0
\(97\) −10.0585 − 4.69035i −1.02128 − 0.476232i −0.161490 − 0.986874i \(-0.551630\pi\)
−0.859794 + 0.510642i \(0.829408\pi\)
\(98\) 0 0
\(99\) −9.60300 − 11.4444i −0.965138 − 1.15021i
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.bh.a.33.5 ✓ 120
5.2 odd 4 inner 380.2.bh.a.337.5 yes 120
19.15 odd 18 inner 380.2.bh.a.53.5 yes 120
95.72 even 36 inner 380.2.bh.a.357.5 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.33.5 ✓ 120 1.1 even 1 trivial
380.2.bh.a.53.5 yes 120 19.15 odd 18 inner
380.2.bh.a.337.5 yes 120 5.2 odd 4 inner
380.2.bh.a.357.5 yes 120 95.72 even 36 inner