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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,2,Mod(7,380)] 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("380.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.v (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [216,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(54\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 7.8
Character \(\chi\) \(=\) 380.7
Dual form 380.2.v.c.163.8

$q$-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+(-1.31685 + 0.515654i) q^{2} +(0.129961 - 0.485021i) q^{3} +(1.46820 - 1.35808i) q^{4} +(2.06245 - 0.863894i) q^{5} +(0.0789637 + 0.705716i) q^{6} +(-2.26251 + 2.26251i) q^{7} +(-1.23310 + 2.54548i) q^{8} +(2.37972 + 1.37393i) q^{9} +(-2.27047 + 2.20113i) q^{10} +1.89777i q^{11} +(-0.467889 - 0.888605i) q^{12} +(0.275985 + 1.02999i) q^{13} +(1.81272 - 4.14607i) q^{14} +(-0.150969 - 1.11260i) q^{15} +(0.311228 - 3.98787i) q^{16} +(-0.590696 + 2.20451i) q^{17} +(-3.84222 - 0.582153i) q^{18} +(4.27916 - 0.829930i) q^{19} +(1.85485 - 4.06934i) q^{20} +(0.803327 + 1.39140i) q^{21} +(-0.978595 - 2.49909i) q^{22} +(4.67925 - 1.25380i) q^{23} +(1.07435 + 0.928893i) q^{24} +(3.50737 - 3.56347i) q^{25} +(-0.894549 - 1.21403i) q^{26} +(2.04084 - 2.04084i) q^{27} +(-0.249146 + 6.39450i) q^{28} +(-3.22266 - 1.86060i) q^{29} +(0.772522 + 1.38728i) q^{30} +0.974031i q^{31} +(1.64652 + 5.41193i) q^{32} +(0.920459 + 0.246636i) q^{33} +(-0.358905 - 3.20761i) q^{34} +(-2.71174 + 6.62088i) q^{35} +(5.35982 - 1.21465i) q^{36} +(-2.22733 - 2.22733i) q^{37} +(-5.20707 + 3.29946i) q^{38} +0.535433 q^{39} +(-0.344186 + 6.31518i) q^{40} +(5.35585 + 9.27661i) q^{41} +(-1.77535 - 1.41803i) q^{42} +(0.544902 - 2.03360i) q^{43} +(2.57733 + 2.78631i) q^{44} +(6.09498 + 0.777835i) q^{45} +(-5.51536 + 4.06395i) q^{46} +(2.40784 + 8.98617i) q^{47} +(-1.89375 - 0.669220i) q^{48} -3.23791i q^{49} +(-2.78117 + 6.50116i) q^{50} +(0.992464 + 0.572999i) q^{51} +(1.80401 + 1.13742i) q^{52} +(-0.980783 - 3.66033i) q^{53} +(-1.63511 + 3.73985i) q^{54} +(1.63947 + 3.91405i) q^{55} +(-2.96926 - 8.54908i) q^{56} +(0.153590 - 2.18334i) q^{57} +(5.20320 + 0.788361i) q^{58} +(-2.52524 - 4.37385i) q^{59} +(-1.73266 - 1.42849i) q^{60} +(6.56975 - 11.3791i) q^{61} +(-0.502264 - 1.28266i) q^{62} +(-8.49268 + 2.27561i) q^{63} +(-4.95891 - 6.27767i) q^{64} +(1.45900 + 1.88588i) q^{65} +(-1.33929 + 0.149855i) q^{66} +(-0.445835 - 1.66388i) q^{67} +(2.12664 + 4.03887i) q^{68} -2.43248i q^{69} +(0.156874 - 10.1170i) q^{70} +(-4.55538 + 2.63005i) q^{71} +(-6.43176 + 4.36333i) q^{72} +(-13.2475 - 3.54966i) q^{73} +(4.08160 + 1.78453i) q^{74} +(-1.27254 - 2.16426i) q^{75} +(5.15556 - 7.02995i) q^{76} +(-4.29373 - 4.29373i) q^{77} +(-0.705086 + 0.276098i) q^{78} +(-5.65657 - 9.79747i) q^{79} +(-2.80321 - 8.49365i) q^{80} +(3.39718 + 5.88409i) q^{81} +(-11.8364 - 9.45415i) q^{82} +(9.42884 + 9.42884i) q^{83} +(3.06908 + 0.951876i) q^{84} +(0.686182 + 5.05698i) q^{85} +(0.331080 + 2.95893i) q^{86} +(-1.32125 + 1.32125i) q^{87} +(-4.83074 - 2.34015i) q^{88} +(-6.74303 - 3.89309i) q^{89} +(-8.42728 + 2.11861i) q^{90} +(-2.95478 - 1.70594i) q^{91} +(5.16732 - 8.19564i) q^{92} +(0.472425 + 0.126586i) q^{93} +(-7.80453 - 10.5919i) q^{94} +(8.10857 - 5.40843i) q^{95} +(2.83888 - 0.0952589i) q^{96} +(3.47557 - 12.9710i) q^{97} +(1.66964 + 4.26385i) q^{98} +(-2.60741 + 4.51617i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 216 q + 12 q^{6} - 36 q^{8} + 4 q^{10} - 20 q^{12} - 8 q^{13} + 8 q^{16} - 16 q^{17} + 12 q^{18} - 48 q^{20} + 40 q^{21} + 16 q^{22} - 16 q^{25} + 24 q^{26} + 8 q^{28} - 12 q^{30} - 20 q^{32} + 20 q^{33}+ \cdots + 34 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.31685 + 0.515654i −0.931155 + 0.364623i
\(3\) 0.129961 − 0.485021i 0.0750330 − 0.280027i −0.918208 − 0.396099i \(-0.870364\pi\)
0.993241 + 0.116072i \(0.0370304\pi\)
\(4\) 1.46820 − 1.35808i 0.734100 − 0.679041i
\(5\) 2.06245 − 0.863894i 0.922354 − 0.386345i
\(6\) 0.0789637 + 0.705716i 0.0322368 + 0.288107i
\(7\) −2.26251 + 2.26251i −0.855149 + 0.855149i −0.990762 − 0.135613i \(-0.956700\pi\)
0.135613 + 0.990762i \(0.456700\pi\)
\(8\) −1.23310 + 2.54548i −0.435968 + 0.899962i
\(9\) 2.37972 + 1.37393i 0.793240 + 0.457978i
\(10\) −2.27047 + 2.20113i −0.717985 + 0.696059i
\(11\) 1.89777i 0.572200i 0.958200 + 0.286100i \(0.0923589\pi\)
−0.958200 + 0.286100i \(0.907641\pi\)
\(12\) −0.467889 − 0.888605i −0.135068 − 0.256518i
\(13\) 0.275985 + 1.02999i 0.0765444 + 0.285667i 0.993579 − 0.113140i \(-0.0360909\pi\)
−0.917035 + 0.398807i \(0.869424\pi\)
\(14\) 1.81272 − 4.14607i 0.484470 − 1.10808i
\(15\) −0.150969 − 1.11260i −0.0389801 − 0.287273i
\(16\) 0.311228 − 3.98787i 0.0778070 − 0.996968i
\(17\) −0.590696 + 2.20451i −0.143265 + 0.534671i 0.856562 + 0.516045i \(0.172596\pi\)
−0.999827 + 0.0186269i \(0.994071\pi\)
\(18\) −3.84222 − 0.582153i −0.905619 − 0.137215i
\(19\) 4.27916 − 0.829930i 0.981707 − 0.190399i
\(20\) 1.85485 − 4.06934i 0.414756 − 0.909932i
\(21\) 0.803327 + 1.39140i 0.175300 + 0.303629i
\(22\) −0.978595 − 2.49909i −0.208637 − 0.532807i
\(23\) 4.67925 − 1.25380i 0.975691 − 0.261436i 0.264462 − 0.964396i \(-0.414806\pi\)
0.711229 + 0.702960i \(0.248139\pi\)
\(24\) 1.07435 + 0.928893i 0.219302 + 0.189610i
\(25\) 3.50737 − 3.56347i 0.701475 − 0.712694i
\(26\) −0.894549 − 1.21403i −0.175436 − 0.238091i
\(27\) 2.04084 − 2.04084i 0.392759 − 0.392759i
\(28\) −0.249146 + 6.39450i −0.0470842 + 1.20845i
\(29\) −3.22266 − 1.86060i −0.598433 − 0.345505i 0.169992 − 0.985445i \(-0.445626\pi\)
−0.768425 + 0.639940i \(0.778959\pi\)
\(30\) 0.772522 + 1.38728i 0.141043 + 0.253282i
\(31\) 0.974031i 0.174941i 0.996167 + 0.0874706i \(0.0278784\pi\)
−0.996167 + 0.0874706i \(0.972122\pi\)
\(32\) 1.64652 + 5.41193i 0.291067 + 0.956703i
\(33\) 0.920459 + 0.246636i 0.160231 + 0.0429338i
\(34\) −0.358905 − 3.20761i −0.0615516 − 0.550100i
\(35\) −2.71174 + 6.62088i −0.458367 + 1.11913i
\(36\) 5.35982 − 1.21465i 0.893304 − 0.202441i
\(37\) −2.22733 − 2.22733i −0.366171 − 0.366171i 0.499908 − 0.866079i \(-0.333367\pi\)
−0.866079 + 0.499908i \(0.833367\pi\)
\(38\) −5.20707 + 3.29946i −0.844698 + 0.535244i
\(39\) 0.535433 0.0857379
\(40\) −0.344186 + 6.31518i −0.0544206 + 0.998518i
\(41\) 5.35585 + 9.27661i 0.836444 + 1.44876i 0.892850 + 0.450355i \(0.148703\pi\)
−0.0564061 + 0.998408i \(0.517964\pi\)
\(42\) −1.77535 − 1.41803i −0.273942 − 0.218807i
\(43\) 0.544902 − 2.03360i 0.0830967 − 0.310121i −0.911850 − 0.410523i \(-0.865346\pi\)
0.994947 + 0.100402i \(0.0320128\pi\)
\(44\) 2.57733 + 2.78631i 0.388547 + 0.420052i
\(45\) 6.09498 + 0.777835i 0.908586 + 0.115953i
\(46\) −5.51536 + 4.06395i −0.813195 + 0.599197i
\(47\) 2.40784 + 8.98617i 0.351219 + 1.31077i 0.885176 + 0.465256i \(0.154038\pi\)
−0.533957 + 0.845512i \(0.679296\pi\)
\(48\) −1.89375 − 0.669220i −0.273340 − 0.0965936i
\(49\) − 3.23791i − 0.462559i
\(50\) −2.78117 + 6.50116i −0.393317 + 0.919403i
\(51\) 0.992464 + 0.572999i 0.138973 + 0.0802360i
\(52\) 1.80401 + 1.13742i 0.250171 + 0.157732i
\(53\) −0.980783 − 3.66033i −0.134721 − 0.502785i −0.999999 − 0.00149297i \(-0.999525\pi\)
0.865278 − 0.501292i \(-0.167142\pi\)
\(54\) −1.63511 + 3.73985i −0.222511 + 0.508929i
\(55\) 1.63947 + 3.91405i 0.221067 + 0.527771i
\(56\) −2.96926 − 8.54908i −0.396784 − 1.14242i
\(57\) 0.153590 − 2.18334i 0.0203435 − 0.289190i
\(58\) 5.20320 + 0.788361i 0.683213 + 0.103517i
\(59\) −2.52524 − 4.37385i −0.328759 − 0.569427i 0.653507 − 0.756920i \(-0.273297\pi\)
−0.982266 + 0.187494i \(0.939964\pi\)
\(60\) −1.73266 − 1.42849i −0.223685 − 0.184418i
\(61\) 6.56975 − 11.3791i 0.841170 − 1.45695i −0.0477367 − 0.998860i \(-0.515201\pi\)
0.888906 − 0.458089i \(-0.151466\pi\)
\(62\) −0.502264 − 1.28266i −0.0637875 − 0.162897i
\(63\) −8.49268 + 2.27561i −1.06998 + 0.286700i
\(64\) −4.95891 − 6.27767i −0.619864 − 0.784709i
\(65\) 1.45900 + 1.88588i 0.180967 + 0.233914i
\(66\) −1.33929 + 0.149855i −0.164855 + 0.0184459i
\(67\) −0.445835 − 1.66388i −0.0544674 − 0.203275i 0.933330 − 0.359020i \(-0.116889\pi\)
−0.987797 + 0.155745i \(0.950222\pi\)
\(68\) 2.12664 + 4.03887i 0.257893 + 0.489785i
\(69\) − 2.43248i − 0.292836i
\(70\) 0.156874 − 10.1170i 0.0187500 − 1.20922i
\(71\) −4.55538 + 2.63005i −0.540624 + 0.312130i −0.745332 − 0.666694i \(-0.767709\pi\)
0.204708 + 0.978823i \(0.434376\pi\)
\(72\) −6.43176 + 4.36333i −0.757990 + 0.514223i
\(73\) −13.2475 − 3.54966i −1.55051 − 0.415457i −0.620863 − 0.783919i \(-0.713218\pi\)
−0.929643 + 0.368463i \(0.879884\pi\)
\(74\) 4.08160 + 1.78453i 0.474476 + 0.207448i
\(75\) −1.27254 − 2.16426i −0.146940 − 0.249907i
\(76\) 5.15556 − 7.02995i 0.591383 − 0.806391i
\(77\) −4.29373 − 4.29373i −0.489316 − 0.489316i
\(78\) −0.705086 + 0.276098i −0.0798353 + 0.0312620i
\(79\) −5.65657 − 9.79747i −0.636414 − 1.10230i −0.986214 − 0.165477i \(-0.947084\pi\)
0.349800 − 0.936824i \(-0.386250\pi\)
\(80\) −2.80321 − 8.49365i −0.313408 − 0.949618i
\(81\) 3.39718 + 5.88409i 0.377464 + 0.653787i
\(82\) −11.8364 − 9.45415i −1.30711 − 1.04404i
\(83\) 9.42884 + 9.42884i 1.03495 + 1.03495i 0.999367 + 0.0355829i \(0.0113288\pi\)
0.0355829 + 0.999367i \(0.488671\pi\)
\(84\) 3.06908 + 0.951876i 0.334865 + 0.103858i
\(85\) 0.686182 + 5.05698i 0.0744269 + 0.548506i
\(86\) 0.331080 + 2.95893i 0.0357013 + 0.319070i
\(87\) −1.32125 + 1.32125i −0.141653 + 0.141653i
\(88\) −4.83074 − 2.34015i −0.514958 − 0.249461i
\(89\) −6.74303 − 3.89309i −0.714760 − 0.412667i 0.0980611 − 0.995180i \(-0.468736\pi\)
−0.812821 + 0.582514i \(0.802069\pi\)
\(90\) −8.42728 + 2.11861i −0.888314 + 0.223321i
\(91\) −2.95478 − 1.70594i −0.309745 − 0.178831i
\(92\) 5.16732 − 8.19564i 0.538730 − 0.854454i
\(93\) 0.472425 + 0.126586i 0.0489882 + 0.0131264i
\(94\) −7.80453 − 10.5919i −0.804975 − 1.09247i
\(95\) 8.10857 − 5.40843i 0.831922 − 0.554893i
\(96\) 2.83888 − 0.0952589i 0.289742 − 0.00972232i
\(97\) 3.47557 − 12.9710i 0.352891 − 1.31701i −0.530227 − 0.847856i \(-0.677893\pi\)
0.883118 − 0.469151i \(-0.155440\pi\)
\(98\) 1.66964 + 4.26385i 0.168660 + 0.430714i
\(99\) −2.60741 + 4.51617i −0.262055 + 0.453892i
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.v.c.7.8 ✓ 216
4.3 odd 2 inner 380.2.v.c.7.39 yes 216
5.3 odd 4 inner 380.2.v.c.83.35 yes 216
19.11 even 3 inner 380.2.v.c.87.43 yes 216
20.3 even 4 inner 380.2.v.c.83.43 yes 216
76.11 odd 6 inner 380.2.v.c.87.35 yes 216
95.68 odd 12 inner 380.2.v.c.163.39 yes 216
380.163 even 12 inner 380.2.v.c.163.8 yes 216
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.v.c.7.8 ✓ 216 1.1 even 1 trivial
380.2.v.c.7.39 yes 216 4.3 odd 2 inner
380.2.v.c.83.35 yes 216 5.3 odd 4 inner
380.2.v.c.83.43 yes 216 20.3 even 4 inner
380.2.v.c.87.35 yes 216 76.11 odd 6 inner
380.2.v.c.87.43 yes 216 19.11 even 3 inner
380.2.v.c.163.8 yes 216 380.163 even 12 inner
380.2.v.c.163.39 yes 216 95.68 odd 12 inner