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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 7.3
Character \(\chi\) \(=\) 380.7
Dual form 380.2.v.c.163.3

$q$-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+(-1.39566 - 0.228298i) q^{2} +(-0.00674551 + 0.0251746i) q^{3} +(1.89576 + 0.637256i) q^{4} +(-2.17162 + 0.532967i) q^{5} +(0.0151618 - 0.0335953i) q^{6} +(-3.02167 + 3.02167i) q^{7} +(-2.50036 - 1.32219i) q^{8} +(2.59749 + 1.49966i) q^{9} +(3.15253 - 0.248066i) q^{10} -5.59169i q^{11} +(-0.0288305 + 0.0434264i) q^{12} +(-1.20316 - 4.49027i) q^{13} +(4.90708 - 3.52739i) q^{14} +(0.00123148 - 0.0582648i) q^{15} +(3.18781 + 2.41617i) q^{16} +(0.369102 - 1.37751i) q^{17} +(-3.28285 - 2.68602i) q^{18} +(-1.63762 - 4.03958i) q^{19} +(-4.45651 - 0.373501i) q^{20} +(-0.0556865 - 0.0964519i) q^{21} +(-1.27657 + 7.80412i) q^{22} +(3.40135 - 0.911389i) q^{23} +(0.0501519 - 0.0540267i) q^{24} +(4.43189 - 2.31481i) q^{25} +(0.654093 + 6.54159i) q^{26} +(-0.110562 + 0.110562i) q^{27} +(-7.65393 + 3.80278i) q^{28} +(-6.80095 - 3.92653i) q^{29} +(-0.0150205 + 0.0810370i) q^{30} +4.44156i q^{31} +(-3.89751 - 4.09993i) q^{32} +(0.140768 + 0.0377188i) q^{33} +(-0.829625 + 1.83827i) q^{34} +(4.95147 - 8.17237i) q^{35} +(3.96855 + 4.49826i) q^{36} +(0.0225777 + 0.0225777i) q^{37} +(1.36333 + 6.01177i) q^{38} +0.121157 q^{39} +(6.13453 + 1.53870i) q^{40} +(-4.62151 - 8.00469i) q^{41} +(0.0556999 + 0.147328i) q^{42} +(-1.31048 + 4.89080i) q^{43} +(3.56334 - 10.6005i) q^{44} +(-6.44003 - 1.87232i) q^{45} +(-4.95521 + 0.495471i) q^{46} +(-0.998551 - 3.72664i) q^{47} +(-0.0823294 + 0.0639535i) q^{48} -11.2609i q^{49} +(-6.71390 + 2.21890i) q^{50} +(0.0321884 + 0.0185840i) q^{51} +(0.580539 - 9.27920i) q^{52} +(-1.82051 - 6.79424i) q^{53} +(0.179548 - 0.129066i) q^{54} +(2.98019 + 12.1430i) q^{55} +(11.5505 - 3.56003i) q^{56} +(0.112741 - 0.0139773i) q^{57} +(8.59543 + 7.03276i) q^{58} +(2.46104 + 4.26265i) q^{59} +(0.0394642 - 0.109671i) q^{60} +(-3.05728 + 5.29537i) q^{61} +(1.01400 - 6.19893i) q^{62} +(-12.3802 + 3.31727i) q^{63} +(4.50361 + 6.61192i) q^{64} +(5.00599 + 9.10993i) q^{65} +(-0.187854 - 0.0847800i) q^{66} +(-1.31656 - 4.91348i) q^{67} +(1.57755 - 2.37621i) q^{68} +0.0917753i q^{69} +(-8.77633 + 10.2755i) q^{70} +(-1.45766 + 0.841583i) q^{71} +(-4.51182 - 7.18407i) q^{72} +(-8.63579 - 2.31395i) q^{73} +(-0.0263565 - 0.0366654i) q^{74} +(0.0283789 + 0.127186i) q^{75} +(-0.530281 - 8.70166i) q^{76} +(16.8962 + 16.8962i) q^{77} +(-0.169094 - 0.0276599i) q^{78} +(0.107623 + 0.186408i) q^{79} +(-8.21046 - 3.54801i) q^{80} +(4.49694 + 7.78893i) q^{81} +(4.62262 + 12.2269i) q^{82} +(1.16162 + 1.16162i) q^{83} +(-0.0441038 - 0.218336i) q^{84} +(-0.0673843 + 3.18815i) q^{85} +(2.94556 - 6.52673i) q^{86} +(0.144725 - 0.144725i) q^{87} +(-7.39330 + 13.9812i) q^{88} +(4.77522 + 2.75698i) q^{89} +(8.56068 + 4.08338i) q^{90} +(17.2037 + 9.93254i) q^{91} +(7.02893 + 0.439754i) q^{92} +(-0.111815 - 0.0299606i) q^{93} +(0.542857 + 5.42911i) q^{94} +(5.70925 + 7.89965i) q^{95} +(0.129505 - 0.0704620i) q^{96} +(-0.845041 + 3.15374i) q^{97} +(-2.57085 + 15.7165i) q^{98} +(8.38564 - 14.5243i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.39566 − 0.228298i −0.986884 − 0.161431i
\(3\) −0.00674551 + 0.0251746i −0.00389452 + 0.0145346i −0.967846 − 0.251544i \(-0.919062\pi\)
0.963951 + 0.266079i \(0.0857282\pi\)
\(4\) 1.89576 + 0.637256i 0.947880 + 0.318628i
\(5\) −2.17162 + 0.532967i −0.971179 + 0.238350i
\(6\) 0.0151618 − 0.0335953i 0.00618977 − 0.0137152i
\(7\) −3.02167 + 3.02167i −1.14208 + 1.14208i −0.154014 + 0.988069i \(0.549220\pi\)
−0.988069 + 0.154014i \(0.950780\pi\)
\(8\) −2.50036 − 1.32219i −0.884011 − 0.467466i
\(9\) 2.59749 + 1.49966i 0.865829 + 0.499887i
\(10\) 3.15253 − 0.248066i 0.996918 − 0.0784453i
\(11\) − 5.59169i − 1.68596i −0.537946 − 0.842979i \(-0.680800\pi\)
0.537946 − 0.842979i \(-0.319200\pi\)
\(12\) −0.0288305 + 0.0434264i −0.00832265 + 0.0125361i
\(13\) −1.20316 − 4.49027i −0.333698 − 1.24538i −0.905274 − 0.424828i \(-0.860335\pi\)
0.571576 − 0.820549i \(-0.306332\pi\)
\(14\) 4.90708 − 3.52739i 1.31147 − 0.942735i
\(15\) 0.00123148 − 0.0582648i 0.000317966 − 0.0150439i
\(16\) 3.18781 + 2.41617i 0.796953 + 0.604042i
\(17\) 0.369102 − 1.37751i 0.0895204 − 0.334095i −0.906611 − 0.421967i \(-0.861340\pi\)
0.996132 + 0.0878719i \(0.0280066\pi\)
\(18\) −3.28285 − 2.68602i −0.773776 − 0.633102i
\(19\) −1.63762 − 4.03958i −0.375695 − 0.926743i
\(20\) −4.45651 − 0.373501i −0.996506 − 0.0835174i
\(21\) −0.0556865 − 0.0964519i −0.0121518 − 0.0210475i
\(22\) −1.27657 + 7.80412i −0.272166 + 1.66384i
\(23\) 3.40135 − 0.911389i 0.709230 − 0.190038i 0.113869 − 0.993496i \(-0.463676\pi\)
0.595361 + 0.803458i \(0.297009\pi\)
\(24\) 0.0501519 − 0.0540267i 0.0102372 − 0.0110281i
\(25\) 4.43189 − 2.31481i 0.886378 − 0.462961i
\(26\) 0.654093 + 6.54159i 0.128278 + 1.28291i
\(27\) −0.110562 + 0.110562i −0.0212777 + 0.0212777i
\(28\) −7.65393 + 3.80278i −1.44646 + 0.718658i
\(29\) −6.80095 − 3.92653i −1.26290 − 0.729138i −0.289269 − 0.957248i \(-0.593412\pi\)
−0.973636 + 0.228109i \(0.926746\pi\)
\(30\) −0.0150205 + 0.0810370i −0.00274235 + 0.0147953i
\(31\) 4.44156i 0.797728i 0.917010 + 0.398864i \(0.130595\pi\)
−0.917010 + 0.398864i \(0.869405\pi\)
\(32\) −3.89751 − 4.09993i −0.688989 − 0.724772i
\(33\) 0.140768 + 0.0377188i 0.0245046 + 0.00656600i
\(34\) −0.829625 + 1.83827i −0.142280 + 0.315261i
\(35\) 4.95147 − 8.17237i 0.836952 − 1.38138i
\(36\) 3.96855 + 4.49826i 0.661424 + 0.749710i
\(37\) 0.0225777 + 0.0225777i 0.00371176 + 0.00371176i 0.708960 − 0.705248i \(-0.249165\pi\)
−0.705248 + 0.708960i \(0.749165\pi\)
\(38\) 1.36333 + 6.01177i 0.221162 + 0.975237i
\(39\) 0.121157 0.0194006
\(40\) 6.13453 + 1.53870i 0.969954 + 0.243289i
\(41\) −4.62151 − 8.00469i −0.721758 − 1.25012i −0.960294 − 0.278989i \(-0.910001\pi\)
0.238536 − 0.971134i \(-0.423332\pi\)
\(42\) 0.0556999 + 0.147328i 0.00859469 + 0.0227331i
\(43\) −1.31048 + 4.89080i −0.199847 + 0.745839i 0.791112 + 0.611672i \(0.209503\pi\)
−0.990959 + 0.134168i \(0.957164\pi\)
\(44\) 3.56334 − 10.6005i 0.537193 − 1.59809i
\(45\) −6.44003 − 1.87232i −0.960024 − 0.279109i
\(46\) −4.95521 + 0.495471i −0.730606 + 0.0730532i
\(47\) −0.998551 − 3.72664i −0.145654 − 0.543587i −0.999725 − 0.0234317i \(-0.992541\pi\)
0.854072 − 0.520155i \(-0.174126\pi\)
\(48\) −0.0823294 + 0.0639535i −0.0118832 + 0.00923090i
\(49\) − 11.2609i − 1.60871i
\(50\) −6.71390 + 2.21890i −0.949489 + 0.313800i
\(51\) 0.0321884 + 0.0185840i 0.00450728 + 0.00260228i
\(52\) 0.580539 − 9.27920i 0.0805063 − 1.28679i
\(53\) −1.82051 − 6.79424i −0.250066 − 0.933261i −0.970769 − 0.240016i \(-0.922847\pi\)
0.720702 − 0.693245i \(-0.243819\pi\)
\(54\) 0.179548 − 0.129066i 0.0244335 − 0.0175637i
\(55\) 2.98019 + 12.1430i 0.401848 + 1.63737i
\(56\) 11.5505 − 3.56003i 1.54350 − 0.475729i
\(57\) 0.112741 − 0.0139773i 0.0149330 − 0.00185133i
\(58\) 8.59543 + 7.03276i 1.12863 + 0.923447i
\(59\) 2.46104 + 4.26265i 0.320400 + 0.554949i 0.980571 − 0.196167i \(-0.0628493\pi\)
−0.660171 + 0.751116i \(0.729516\pi\)
\(60\) 0.0394642 − 0.109671i 0.00509480 − 0.0141585i
\(61\) −3.05728 + 5.29537i −0.391445 + 0.678003i −0.992640 − 0.121099i \(-0.961358\pi\)
0.601195 + 0.799102i \(0.294691\pi\)
\(62\) 1.01400 − 6.19893i 0.128778 − 0.787265i
\(63\) −12.3802 + 3.31727i −1.55976 + 0.417937i
\(64\) 4.50361 + 6.61192i 0.562951 + 0.826490i
\(65\) 5.00599 + 9.10993i 0.620916 + 1.12995i
\(66\) −0.187854 − 0.0847800i −0.0231233 − 0.0104357i
\(67\) −1.31656 − 4.91348i −0.160844 − 0.600277i −0.998534 − 0.0541310i \(-0.982761\pi\)
0.837690 − 0.546146i \(-0.183906\pi\)
\(68\) 1.57755 − 2.37621i 0.191306 − 0.288158i
\(69\) 0.0917753i 0.0110485i
\(70\) −8.77633 + 10.2755i −1.04897 + 1.22815i
\(71\) −1.45766 + 0.841583i −0.172993 + 0.0998776i −0.583996 − 0.811756i \(-0.698512\pi\)
0.411003 + 0.911634i \(0.365178\pi\)
\(72\) −4.51182 − 7.18407i −0.531723 − 0.846651i
\(73\) −8.63579 − 2.31395i −1.01074 − 0.270828i −0.284803 − 0.958586i \(-0.591928\pi\)
−0.725940 + 0.687758i \(0.758595\pi\)
\(74\) −0.0263565 − 0.0366654i −0.00306388 − 0.00426227i
\(75\) 0.0283789 + 0.127186i 0.00327692 + 0.0146861i
\(76\) −0.530281 − 8.70166i −0.0608274 − 0.998148i
\(77\) 16.8962 + 16.8962i 1.92550 + 1.92550i
\(78\) −0.169094 − 0.0276599i −0.0191461 − 0.00313186i
\(79\) 0.107623 + 0.186408i 0.0121085 + 0.0209725i 0.872016 − 0.489477i \(-0.162812\pi\)
−0.859908 + 0.510450i \(0.829479\pi\)
\(80\) −8.21046 − 3.54801i −0.917957 − 0.396679i
\(81\) 4.49694 + 7.78893i 0.499660 + 0.865437i
\(82\) 4.62262 + 12.2269i 0.510483 + 1.35024i
\(83\) 1.16162 + 1.16162i 0.127505 + 0.127505i 0.767979 − 0.640475i \(-0.221262\pi\)
−0.640475 + 0.767979i \(0.721262\pi\)
\(84\) −0.0441038 − 0.218336i −0.00481212 − 0.0238224i
\(85\) −0.0673843 + 3.18815i −0.00730885 + 0.345803i
\(86\) 2.94556 − 6.52673i 0.317628 − 0.703795i
\(87\) 0.144725 − 0.144725i 0.0155161 − 0.0155161i
\(88\) −7.39330 + 13.9812i −0.788128 + 1.49041i
\(89\) 4.77522 + 2.75698i 0.506173 + 0.292239i 0.731259 − 0.682100i \(-0.238933\pi\)
−0.225086 + 0.974339i \(0.572266\pi\)
\(90\) 8.56068 + 4.08338i 0.902375 + 0.430426i
\(91\) 17.2037 + 9.93254i 1.80343 + 1.04121i
\(92\) 7.02893 + 0.439754i 0.732816 + 0.0458476i
\(93\) −0.111815 − 0.0299606i −0.0115946 − 0.00310677i
\(94\) 0.542857 + 5.42911i 0.0559914 + 0.559970i
\(95\) 5.70925 + 7.89965i 0.585756 + 0.810487i
\(96\) 0.129505 − 0.0704620i 0.0132175 − 0.00719150i
\(97\) −0.845041 + 3.15374i −0.0858009 + 0.320213i −0.995465 − 0.0951321i \(-0.969673\pi\)
0.909664 + 0.415345i \(0.136339\pi\)
\(98\) −2.57085 + 15.7165i −0.259695 + 1.58761i
\(99\) 8.38564 − 14.5243i 0.842788 − 1.45975i
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.7.3 ✓ 216
4.3 odd 2 inner 380.2.v.c.7.48 yes 216
5.3 odd 4 inner 380.2.v.c.83.25 yes 216
19.11 even 3 inner 380.2.v.c.87.34 yes 216
20.3 even 4 inner 380.2.v.c.83.34 yes 216
76.11 odd 6 inner 380.2.v.c.87.25 yes 216
95.68 odd 12 inner 380.2.v.c.163.48 yes 216
380.163 even 12 inner 380.2.v.c.163.3 yes 216
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.v.c.7.3 ✓ 216 1.1 even 1 trivial
380.2.v.c.7.48 yes 216 4.3 odd 2 inner
380.2.v.c.83.25 yes 216 5.3 odd 4 inner
380.2.v.c.83.34 yes 216 20.3 even 4 inner
380.2.v.c.87.25 yes 216 76.11 odd 6 inner
380.2.v.c.87.34 yes 216 19.11 even 3 inner
380.2.v.c.163.3 yes 216 380.163 even 12 inner
380.2.v.c.163.48 yes 216 95.68 odd 12 inner