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Properties

Label 380.3.h.a.39.15
Level $380$
Weight $3$
Character 380.39
Analytic conductor $10.354$
Analytic rank $0$
Dimension $108$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(39,380)] 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("380.39"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.h (of order \(2\), degree \(1\), 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: \(10.3542500457\)
Analytic rank: \(0\)
Dimension: \(108\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 39.15
Character \(\chi\) \(=\) 380.39
Dual form 380.3.h.a.39.16

$q$-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+(-1.75001 - 0.968228i) q^{2} -4.24708 q^{3} +(2.12507 + 3.38882i) q^{4} +(4.77157 + 1.49403i) q^{5} +(7.43242 + 4.11214i) q^{6} +7.86306 q^{7} +(-0.437742 - 7.98801i) q^{8} +9.03765 q^{9} +(-6.90374 - 7.23453i) q^{10} +2.63701i q^{11} +(-9.02533 - 14.3926i) q^{12} -5.11257i q^{13} +(-13.7604 - 7.61323i) q^{14} +(-20.2652 - 6.34524i) q^{15} +(-6.96817 + 14.4029i) q^{16} +12.2680i q^{17} +(-15.8160 - 8.75051i) q^{18} +4.35890i q^{19} +(5.07693 + 19.3449i) q^{20} -33.3950 q^{21} +(2.55323 - 4.61480i) q^{22} -17.2507 q^{23} +(1.85912 + 33.9257i) q^{24} +(20.5358 + 14.2577i) q^{25} +(-4.95014 + 8.94705i) q^{26} -0.159903 q^{27} +(16.7095 + 26.6465i) q^{28} +4.95966 q^{29} +(29.3207 + 30.7256i) q^{30} -59.4651i q^{31} +(26.1397 - 18.4585i) q^{32} -11.1996i q^{33} +(11.8783 - 21.4692i) q^{34} +(37.5191 + 11.7476i) q^{35} +(19.2056 + 30.6269i) q^{36} +47.8677i q^{37} +(4.22041 - 7.62812i) q^{38} +21.7135i q^{39} +(9.84558 - 38.7694i) q^{40} -4.38513 q^{41} +(58.4416 + 32.3340i) q^{42} +51.2362 q^{43} +(-8.93636 + 5.60383i) q^{44} +(43.1238 + 13.5025i) q^{45} +(30.1889 + 16.7026i) q^{46} +41.3609 q^{47} +(29.5943 - 61.1704i) q^{48} +12.8277 q^{49} +(-22.1331 - 44.8344i) q^{50} -52.1032i q^{51} +(17.3256 - 10.8646i) q^{52} +42.9820i q^{53} +(0.279832 + 0.154823i) q^{54} +(-3.93976 + 12.5827i) q^{55} +(-3.44199 - 62.8102i) q^{56} -18.5126i q^{57} +(-8.67945 - 4.80208i) q^{58} +103.336i q^{59} +(-21.5621 - 82.1592i) q^{60} +68.7279 q^{61} +(-57.5757 + 104.064i) q^{62} +71.0636 q^{63} +(-63.6168 + 6.99338i) q^{64} +(7.63831 - 24.3950i) q^{65} +(-10.8438 + 19.5994i) q^{66} +69.7930 q^{67} +(-41.5741 + 26.0704i) q^{68} +73.2652 q^{69} +(-54.2845 - 56.8855i) q^{70} +55.1475i q^{71} +(-3.95616 - 72.1929i) q^{72} -45.6447i q^{73} +(46.3468 - 83.7689i) q^{74} +(-87.2170 - 60.5535i) q^{75} +(-14.7715 + 9.26296i) q^{76} +20.7350i q^{77} +(21.0236 - 37.9988i) q^{78} +84.1513i q^{79} +(-54.7675 + 58.3140i) q^{80} -80.6597 q^{81} +(7.67402 + 4.24580i) q^{82} -2.16220 q^{83} +(-70.9667 - 113.170i) q^{84} +(-18.3287 + 58.5378i) q^{85} +(-89.6639 - 49.6084i) q^{86} -21.0640 q^{87} +(21.0645 - 1.15433i) q^{88} +89.6522 q^{89} +(-62.3936 - 65.3831i) q^{90} -40.2004i q^{91} +(-36.6590 - 58.4596i) q^{92} +252.553i q^{93} +(-72.3821 - 40.0468i) q^{94} +(-6.51231 + 20.7988i) q^{95} +(-111.017 + 78.3947i) q^{96} +36.5447i q^{97} +(-22.4485 - 12.4201i) q^{98} +23.8324i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 108 q + 4 q^{5} + 324 q^{9} - 8 q^{10} + 8 q^{14} - 104 q^{16} - 16 q^{21} - 8 q^{24} - 76 q^{25} + 80 q^{26} - 88 q^{29} - 140 q^{30} - 88 q^{34} - 256 q^{36} + 44 q^{40} - 200 q^{41} - 8 q^{44} + 108 q^{45}+ \cdots + 720 q^{96}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.75001 − 0.968228i −0.875005 − 0.484114i
\(3\) −4.24708 −1.41569 −0.707846 − 0.706367i \(-0.750333\pi\)
−0.707846 + 0.706367i \(0.750333\pi\)
\(4\) 2.12507 + 3.38882i 0.531267 + 0.847204i
\(5\) 4.77157 + 1.49403i 0.954314 + 0.298805i
\(6\) 7.43242 + 4.11214i 1.23874 + 0.685356i
\(7\) 7.86306 1.12329 0.561647 − 0.827377i \(-0.310168\pi\)
0.561647 + 0.827377i \(0.310168\pi\)
\(8\) −0.437742 − 7.98801i −0.0547178 − 0.998502i
\(9\) 9.03765 1.00418
\(10\) −6.90374 − 7.23453i −0.690374 − 0.723453i
\(11\) 2.63701i 0.239728i 0.992790 + 0.119864i \(0.0382459\pi\)
−0.992790 + 0.119864i \(0.961754\pi\)
\(12\) −9.02533 − 14.3926i −0.752111 − 1.19938i
\(13\) − 5.11257i − 0.393275i −0.980476 − 0.196637i \(-0.936998\pi\)
0.980476 − 0.196637i \(-0.0630022\pi\)
\(14\) −13.7604 − 7.61323i −0.982888 − 0.543802i
\(15\) −20.2652 − 6.34524i −1.35101 − 0.423016i
\(16\) −6.96817 + 14.4029i −0.435511 + 0.900184i
\(17\) 12.2680i 0.721649i 0.932634 + 0.360824i \(0.117505\pi\)
−0.932634 + 0.360824i \(0.882495\pi\)
\(18\) −15.8160 − 8.75051i −0.878665 − 0.486139i
\(19\) 4.35890i 0.229416i
\(20\) 5.07693 + 19.3449i 0.253847 + 0.967244i
\(21\) −33.3950 −1.59024
\(22\) 2.55323 − 4.61480i 0.116056 − 0.209764i
\(23\) −17.2507 −0.750032 −0.375016 − 0.927018i \(-0.622363\pi\)
−0.375016 + 0.927018i \(0.622363\pi\)
\(24\) 1.85912 + 33.9257i 0.0774635 + 1.41357i
\(25\) 20.5358 + 14.2577i 0.821431 + 0.570308i
\(26\) −4.95014 + 8.94705i −0.190390 + 0.344117i
\(27\) −0.159903 −0.00592234
\(28\) 16.7095 + 26.6465i 0.596769 + 0.951659i
\(29\) 4.95966 0.171023 0.0855114 − 0.996337i \(-0.472748\pi\)
0.0855114 + 0.996337i \(0.472748\pi\)
\(30\) 29.3207 + 30.7256i 0.977357 + 1.02419i
\(31\) − 59.4651i − 1.91823i −0.283020 − 0.959114i \(-0.591336\pi\)
0.283020 − 0.959114i \(-0.408664\pi\)
\(32\) 26.1397 − 18.4585i 0.816865 − 0.576828i
\(33\) − 11.1996i − 0.339382i
\(34\) 11.8783 − 21.4692i 0.349360 − 0.631446i
\(35\) 37.5191 + 11.7476i 1.07198 + 0.335646i
\(36\) 19.2056 + 30.6269i 0.533490 + 0.850749i
\(37\) 47.8677i 1.29372i 0.762609 + 0.646860i \(0.223918\pi\)
−0.762609 + 0.646860i \(0.776082\pi\)
\(38\) 4.22041 − 7.62812i 0.111063 − 0.200740i
\(39\) 21.7135i 0.556756i
\(40\) 9.84558 − 38.7694i 0.246140 − 0.969234i
\(41\) −4.38513 −0.106954 −0.0534772 − 0.998569i \(-0.517030\pi\)
−0.0534772 + 0.998569i \(0.517030\pi\)
\(42\) 58.4416 + 32.3340i 1.39147 + 0.769857i
\(43\) 51.2362 1.19154 0.595770 − 0.803155i \(-0.296847\pi\)
0.595770 + 0.803155i \(0.296847\pi\)
\(44\) −8.93636 + 5.60383i −0.203099 + 0.127360i
\(45\) 43.1238 + 13.5025i 0.958306 + 0.300055i
\(46\) 30.1889 + 16.7026i 0.656281 + 0.363101i
\(47\) 41.3609 0.880020 0.440010 − 0.897993i \(-0.354975\pi\)
0.440010 + 0.897993i \(0.354975\pi\)
\(48\) 29.5943 − 61.1704i 0.616549 − 1.27438i
\(49\) 12.8277 0.261789
\(50\) −22.1331 − 44.8344i −0.442662 − 0.896689i
\(51\) − 52.1032i − 1.02163i
\(52\) 17.3256 − 10.8646i 0.333184 − 0.208934i
\(53\) 42.9820i 0.810982i 0.914099 + 0.405491i \(0.132899\pi\)
−0.914099 + 0.405491i \(0.867101\pi\)
\(54\) 0.279832 + 0.154823i 0.00518208 + 0.00286709i
\(55\) −3.93976 + 12.5827i −0.0716321 + 0.228776i
\(56\) −3.44199 − 62.8102i −0.0614641 − 1.12161i
\(57\) − 18.5126i − 0.324782i
\(58\) −8.67945 − 4.80208i −0.149646 − 0.0827945i
\(59\) 103.336i 1.75146i 0.482800 + 0.875731i \(0.339620\pi\)
−0.482800 + 0.875731i \(0.660380\pi\)
\(60\) −21.5621 − 82.1592i −0.359369 − 1.36932i
\(61\) 68.7279 1.12669 0.563343 − 0.826223i \(-0.309515\pi\)
0.563343 + 0.826223i \(0.309515\pi\)
\(62\) −57.5757 + 104.064i −0.928641 + 1.67846i
\(63\) 71.0636 1.12799
\(64\) −63.6168 + 6.99338i −0.994012 + 0.109272i
\(65\) 7.63831 − 24.3950i 0.117513 − 0.375308i
\(66\) −10.8438 + 19.5994i −0.164299 + 0.296961i
\(67\) 69.7930 1.04169 0.520843 − 0.853652i \(-0.325618\pi\)
0.520843 + 0.853652i \(0.325618\pi\)
\(68\) −41.5741 + 26.0704i −0.611384 + 0.383388i
\(69\) 73.2652 1.06181
\(70\) −54.2845 − 56.8855i −0.775493 − 0.812650i
\(71\) 55.1475i 0.776725i 0.921507 + 0.388362i \(0.126959\pi\)
−0.921507 + 0.388362i \(0.873041\pi\)
\(72\) −3.95616 − 72.1929i −0.0549467 − 1.00268i
\(73\) − 45.6447i − 0.625270i −0.949873 − 0.312635i \(-0.898788\pi\)
0.949873 − 0.312635i \(-0.101212\pi\)
\(74\) 46.3468 − 83.7689i 0.626308 − 1.13201i
\(75\) −87.2170 − 60.5535i −1.16289 − 0.807380i
\(76\) −14.7715 + 9.26296i −0.194362 + 0.121881i
\(77\) 20.7350i 0.269285i
\(78\) 21.0236 − 37.9988i 0.269533 − 0.487164i
\(79\) 84.1513i 1.06521i 0.846365 + 0.532603i \(0.178786\pi\)
−0.846365 + 0.532603i \(0.821214\pi\)
\(80\) −54.7675 + 58.3140i −0.684593 + 0.728925i
\(81\) −80.6597 −0.995799
\(82\) 7.67402 + 4.24580i 0.0935856 + 0.0517781i
\(83\) −2.16220 −0.0260506 −0.0130253 − 0.999915i \(-0.504146\pi\)
−0.0130253 + 0.999915i \(0.504146\pi\)
\(84\) −70.9667 − 113.170i −0.844841 − 1.34726i
\(85\) −18.3287 + 58.5378i −0.215632 + 0.688680i
\(86\) −89.6639 − 49.6084i −1.04260 − 0.576842i
\(87\) −21.0640 −0.242115
\(88\) 21.0645 − 1.15433i 0.239369 − 0.0131174i
\(89\) 89.6522 1.00733 0.503664 − 0.863900i \(-0.331985\pi\)
0.503664 + 0.863900i \(0.331985\pi\)
\(90\) −62.3936 − 65.3831i −0.693262 − 0.726479i
\(91\) − 40.2004i − 0.441763i
\(92\) −36.6590 − 58.4596i −0.398467 − 0.635430i
\(93\) 252.553i 2.71562i
\(94\) −72.3821 − 40.0468i −0.770022 − 0.426030i
\(95\) −6.51231 + 20.7988i −0.0685506 + 0.218935i
\(96\) −111.017 + 78.3947i −1.15643 + 0.816611i
\(97\) 36.5447i 0.376750i 0.982097 + 0.188375i \(0.0603220\pi\)
−0.982097 + 0.188375i \(0.939678\pi\)
\(98\) −22.4485 − 12.4201i −0.229067 − 0.126736i
\(99\) 23.8324i 0.240731i
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 380.3.h.a.39.15 ✓ 108
4.3 odd 2 inner 380.3.h.a.39.93 yes 108
5.4 even 2 inner 380.3.h.a.39.94 yes 108
20.19 odd 2 inner 380.3.h.a.39.16 yes 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.h.a.39.15 ✓ 108 1.1 even 1 trivial
380.3.h.a.39.16 yes 108 20.19 odd 2 inner
380.3.h.a.39.93 yes 108 4.3 odd 2 inner
380.3.h.a.39.94 yes 108 5.4 even 2 inner