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Properties

Label 380.3.j.a.227.19
Level $380$
Weight $3$
Character 380.227
Analytic conductor $10.354$
Analytic rank $0$
Dimension $232$
Inner twists $8$

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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(227,380)] 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("380.227"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 2])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.j (of order \(4\), degree \(2\), 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: \(232\)
Relative dimension: \(116\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 227.19
Character \(\chi\) \(=\) 380.227
Dual form 380.3.j.a.303.19

$q$-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+(-1.74392 - 0.979150i) q^{2} +(3.79260 - 3.79260i) q^{3} +(2.08253 + 3.41512i) q^{4} +(-2.62716 - 4.25418i) q^{5} +(-10.3275 + 2.90047i) q^{6} +(8.90960 - 8.90960i) q^{7} +(-0.287859 - 7.99482i) q^{8} -19.7676i q^{9} +(0.416087 + 9.99134i) q^{10} -7.90740i q^{11} +(20.8504 + 5.05398i) q^{12} +(14.5048 + 14.5048i) q^{13} +(-24.2615 + 6.81382i) q^{14} +(-26.0981 - 6.17062i) q^{15} +(-7.32612 + 14.2242i) q^{16} +(3.69193 + 3.69193i) q^{17} +(-19.3554 + 34.4731i) q^{18} +(7.37304 + 17.5111i) q^{19} +(9.05739 - 17.8315i) q^{20} -67.5811i q^{21} +(-7.74253 + 13.7899i) q^{22} +(2.80581 + 2.80581i) q^{23} +(-31.4129 - 29.2294i) q^{24} +(-11.1961 + 22.3528i) q^{25} +(-11.0929 - 39.4976i) q^{26} +(-40.8370 - 40.8370i) q^{27} +(48.9819 + 11.8728i) q^{28} +25.3533 q^{29} +(39.4712 + 36.3151i) q^{30} -5.91985 q^{31} +(26.7038 - 17.6325i) q^{32} +(-29.9896 - 29.9896i) q^{33} +(-2.82349 - 10.0534i) q^{34} +(-61.3100 - 14.4961i) q^{35} +(67.5087 - 41.1666i) q^{36} +(-26.9077 + 26.9077i) q^{37} +(4.28796 - 37.7573i) q^{38} +110.022 q^{39} +(-33.2551 + 22.2283i) q^{40} +52.3286i q^{41} +(-66.1720 + 117.856i) q^{42} +(23.5431 + 23.5431i) q^{43} +(27.0047 - 16.4674i) q^{44} +(-84.0947 + 51.9326i) q^{45} +(-2.14581 - 7.64042i) q^{46} +(-23.0872 + 23.0872i) q^{47} +(26.1616 + 81.7317i) q^{48} -109.762i q^{49} +(41.4118 - 28.0190i) q^{50} +28.0040 q^{51} +(-19.3289 + 79.7423i) q^{52} +(-21.6352 - 21.6352i) q^{53} +(31.2311 + 111.202i) q^{54} +(-33.6395 + 20.7740i) q^{55} +(-73.7954 - 68.6660i) q^{56} +(94.3754 + 38.4495i) q^{57} +(-44.2141 - 24.8246i) q^{58} +31.2486i q^{59} +(-33.2768 - 101.979i) q^{60} -81.6686 q^{61} +(10.3238 + 5.79642i) q^{62} +(-176.121 - 176.121i) q^{63} +(-63.8343 + 4.60277i) q^{64} +(23.5996 - 99.8124i) q^{65} +(22.9352 + 81.6638i) q^{66} +(16.1132 + 16.1132i) q^{67} +(-4.91983 + 20.2970i) q^{68} +21.2826 q^{69} +(92.7261 + 85.3117i) q^{70} -53.6849 q^{71} +(-158.038 + 5.69028i) q^{72} +(39.1260 - 39.1260i) q^{73} +(73.2716 - 20.5783i) q^{74} +(42.3131 + 127.237i) q^{75} +(-44.4479 + 61.6472i) q^{76} +(-70.4518 - 70.4518i) q^{77} +(-191.869 - 107.728i) q^{78} +47.4171i q^{79} +(79.7592 - 6.20264i) q^{80} -131.849 q^{81} +(51.2375 - 91.2570i) q^{82} +(69.0852 + 69.0852i) q^{83} +(230.798 - 140.740i) q^{84} +(6.00684 - 25.4054i) q^{85} +(-18.0051 - 64.1095i) q^{86} +(96.1547 - 96.1547i) q^{87} +(-63.2182 + 2.27622i) q^{88} -31.8283 q^{89} +(197.504 - 8.22503i) q^{90} +258.464 q^{91} +(-3.73899 + 15.4254i) q^{92} +(-22.4516 + 22.4516i) q^{93} +(62.8682 - 17.6565i) q^{94} +(55.1251 - 77.3707i) q^{95} +(34.4036 - 168.150i) q^{96} +(-85.0514 + 85.0514i) q^{97} +(-107.473 + 191.417i) q^{98} -156.310 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 232 q - 8 q^{5} - 32 q^{6} - 64 q^{16} - 8 q^{17} + 80 q^{20} - 8 q^{25} + 40 q^{26} + 40 q^{28} - 104 q^{30} + 104 q^{36} + 184 q^{38} + 192 q^{42} + 192 q^{45} + 280 q^{58} + 112 q^{61} - 280 q^{62} + 216 q^{66}+ \cdots + 504 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.74392 − 0.979150i −0.871961 − 0.489575i
\(3\) 3.79260 − 3.79260i 1.26420 − 1.26420i 0.315160 − 0.949039i \(-0.397942\pi\)
0.949039 − 0.315160i \(-0.102058\pi\)
\(4\) 2.08253 + 3.41512i 0.520633 + 0.853781i
\(5\) −2.62716 − 4.25418i −0.525432 − 0.850836i
\(6\) −10.3275 + 2.90047i −1.72125 + 0.483412i
\(7\) 8.90960 − 8.90960i 1.27280 − 1.27280i 0.328188 − 0.944612i \(-0.393562\pi\)
0.944612 − 0.328188i \(-0.106438\pi\)
\(8\) −0.287859 − 7.99482i −0.0359824 − 0.999352i
\(9\) − 19.7676i − 2.19640i
\(10\) 0.416087 + 9.99134i 0.0416087 + 0.999134i
\(11\) − 7.90740i − 0.718855i −0.933173 − 0.359427i \(-0.882972\pi\)
0.933173 − 0.359427i \(-0.117028\pi\)
\(12\) 20.8504 + 5.05398i 1.73753 + 0.421165i
\(13\) 14.5048 + 14.5048i 1.11575 + 1.11575i 0.992358 + 0.123395i \(0.0393783\pi\)
0.123395 + 0.992358i \(0.460622\pi\)
\(14\) −24.2615 + 6.81382i −1.73296 + 0.486702i
\(15\) −26.0981 − 6.17062i −1.73988 − 0.411375i
\(16\) −7.32612 + 14.2242i −0.457882 + 0.889013i
\(17\) 3.69193 + 3.69193i 0.217173 + 0.217173i 0.807306 − 0.590133i \(-0.200925\pi\)
−0.590133 + 0.807306i \(0.700925\pi\)
\(18\) −19.3554 + 34.4731i −1.07530 + 1.91517i
\(19\) 7.37304 + 17.5111i 0.388055 + 0.921636i
\(20\) 9.05739 − 17.8315i 0.452870 − 0.891577i
\(21\) − 67.5811i − 3.21815i
\(22\) −7.74253 + 13.7899i −0.351933 + 0.626813i
\(23\) 2.80581 + 2.80581i 0.121992 + 0.121992i 0.765467 − 0.643475i \(-0.222508\pi\)
−0.643475 + 0.765467i \(0.722508\pi\)
\(24\) −31.4129 − 29.2294i −1.30887 − 1.21789i
\(25\) −11.1961 + 22.3528i −0.447842 + 0.894113i
\(26\) −11.0929 − 39.4976i −0.426649 − 1.51914i
\(27\) −40.8370 − 40.8370i −1.51248 − 1.51248i
\(28\) 48.9819 + 11.8728i 1.74935 + 0.424030i
\(29\) 25.3533 0.874251 0.437125 − 0.899401i \(-0.355997\pi\)
0.437125 + 0.899401i \(0.355997\pi\)
\(30\) 39.4712 + 36.3151i 1.31571 + 1.21050i
\(31\) −5.91985 −0.190963 −0.0954815 − 0.995431i \(-0.530439\pi\)
−0.0954815 + 0.995431i \(0.530439\pi\)
\(32\) 26.7038 − 17.6325i 0.834494 − 0.551017i
\(33\) −29.9896 − 29.9896i −0.908775 − 0.908775i
\(34\) −2.82349 − 10.0534i −0.0830439 − 0.295688i
\(35\) −61.3100 − 14.4961i −1.75171 − 0.414174i
\(36\) 67.5087 − 41.1666i 1.87524 − 1.14352i
\(37\) −26.9077 + 26.9077i −0.727235 + 0.727235i −0.970068 − 0.242833i \(-0.921923\pi\)
0.242833 + 0.970068i \(0.421923\pi\)
\(38\) 4.28796 − 37.7573i 0.112841 − 0.993613i
\(39\) 110.022 2.82107
\(40\) −33.2551 + 22.2283i −0.831378 + 0.555707i
\(41\) 52.3286i 1.27631i 0.769909 + 0.638154i \(0.220302\pi\)
−0.769909 + 0.638154i \(0.779698\pi\)
\(42\) −66.1720 + 117.856i −1.57552 + 2.80610i
\(43\) 23.5431 + 23.5431i 0.547514 + 0.547514i 0.925721 − 0.378207i \(-0.123459\pi\)
−0.378207 + 0.925721i \(0.623459\pi\)
\(44\) 27.0047 − 16.4674i 0.613744 − 0.374259i
\(45\) −84.0947 + 51.9326i −1.86877 + 1.15406i
\(46\) −2.14581 − 7.64042i −0.0466480 − 0.166096i
\(47\) −23.0872 + 23.0872i −0.491218 + 0.491218i −0.908690 − 0.417472i \(-0.862916\pi\)
0.417472 + 0.908690i \(0.362916\pi\)
\(48\) 26.1616 + 81.7317i 0.545034 + 1.70274i
\(49\) − 109.762i − 2.24004i
\(50\) 41.4118 − 28.0190i 0.828236 − 0.560379i
\(51\) 28.0040 0.549098
\(52\) −19.3289 + 79.7423i −0.371710 + 1.53351i
\(53\) −21.6352 − 21.6352i −0.408211 − 0.408211i 0.472903 − 0.881114i \(-0.343206\pi\)
−0.881114 + 0.472903i \(0.843206\pi\)
\(54\) 31.2311 + 111.202i 0.578353 + 2.05930i
\(55\) −33.6395 + 20.7740i −0.611627 + 0.377709i
\(56\) −73.7954 − 68.6660i −1.31777 − 1.22618i
\(57\) 94.3754 + 38.4495i 1.65571 + 0.674553i
\(58\) −44.2141 − 24.8246i −0.762313 − 0.428011i
\(59\) 31.2486i 0.529637i 0.964298 + 0.264818i \(0.0853120\pi\)
−0.964298 + 0.264818i \(0.914688\pi\)
\(60\) −33.2768 − 101.979i −0.554613 − 1.69965i
\(61\) −81.6686 −1.33883 −0.669414 − 0.742889i \(-0.733455\pi\)
−0.669414 + 0.742889i \(0.733455\pi\)
\(62\) 10.3238 + 5.79642i 0.166512 + 0.0934907i
\(63\) −176.121 − 176.121i −2.79557 − 2.79557i
\(64\) −63.8343 + 4.60277i −0.997411 + 0.0719182i
\(65\) 23.5996 − 99.8124i 0.363070 − 1.53557i
\(66\) 22.9352 + 81.6638i 0.347503 + 1.23733i
\(67\) 16.1132 + 16.1132i 0.240496 + 0.240496i 0.817055 − 0.576559i \(-0.195605\pi\)
−0.576559 + 0.817055i \(0.695605\pi\)
\(68\) −4.91983 + 20.2970i −0.0723505 + 0.298485i
\(69\) 21.2826 0.308444
\(70\) 92.7261 + 85.3117i 1.32466 + 1.21874i
\(71\) −53.6849 −0.756126 −0.378063 − 0.925780i \(-0.623410\pi\)
−0.378063 + 0.925780i \(0.623410\pi\)
\(72\) −158.038 + 5.69028i −2.19497 + 0.0790317i
\(73\) 39.1260 − 39.1260i 0.535972 − 0.535972i −0.386371 − 0.922343i \(-0.626272\pi\)
0.922343 + 0.386371i \(0.126272\pi\)
\(74\) 73.2716 − 20.5783i 0.990156 − 0.278085i
\(75\) 42.3131 + 127.237i 0.564174 + 1.69650i
\(76\) −44.4479 + 61.6472i −0.584841 + 0.811148i
\(77\) −70.4518 − 70.4518i −0.914959 − 0.914959i
\(78\) −191.869 − 107.728i −2.45986 − 1.38112i
\(79\) 47.4171i 0.600216i 0.953905 + 0.300108i \(0.0970227\pi\)
−0.953905 + 0.300108i \(0.902977\pi\)
\(80\) 79.7592 − 6.20264i 0.996990 − 0.0775330i
\(81\) −131.849 −1.62776
\(82\) 51.2375 − 91.2570i 0.624848 − 1.11289i
\(83\) 69.0852 + 69.0852i 0.832352 + 0.832352i 0.987838 − 0.155486i \(-0.0496945\pi\)
−0.155486 + 0.987838i \(0.549695\pi\)
\(84\) 230.798 − 140.740i 2.74759 − 1.67547i
\(85\) 6.00684 − 25.4054i 0.0706687 − 0.298888i
\(86\) −18.0051 − 64.1095i −0.209362 − 0.745460i
\(87\) 96.1547 − 96.1547i 1.10523 − 1.10523i
\(88\) −63.2182 + 2.27622i −0.718389 + 0.0258661i
\(89\) −31.8283 −0.357621 −0.178810 − 0.983884i \(-0.557225\pi\)
−0.178810 + 0.983884i \(0.557225\pi\)
\(90\) 197.504 − 8.22503i 2.19449 − 0.0913892i
\(91\) 258.464 2.84026
\(92\) −3.73899 + 15.4254i −0.0406412 + 0.167667i
\(93\) −22.4516 + 22.4516i −0.241415 + 0.241415i
\(94\) 62.8682 − 17.6565i 0.668811 − 0.187835i
\(95\) 55.1251 − 77.3707i 0.580265 − 0.814428i
\(96\) 34.4036 − 168.150i 0.358371 − 1.75156i
\(97\) −85.0514 + 85.0514i −0.876818 + 0.876818i −0.993204 − 0.116386i \(-0.962869\pi\)
0.116386 + 0.993204i \(0.462869\pi\)
\(98\) −107.473 + 191.417i −1.09667 + 1.95323i
\(99\) −156.310 −1.57889
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.j.a.227.19 ✓ 232
4.3 odd 2 inner 380.3.j.a.227.77 yes 232
5.3 odd 4 inner 380.3.j.a.303.40 yes 232
19.18 odd 2 inner 380.3.j.a.227.98 yes 232
20.3 even 4 inner 380.3.j.a.303.98 yes 232
76.75 even 2 inner 380.3.j.a.227.40 yes 232
95.18 even 4 inner 380.3.j.a.303.77 yes 232
380.303 odd 4 inner 380.3.j.a.303.19 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.19 ✓ 232 1.1 even 1 trivial
380.3.j.a.227.40 yes 232 76.75 even 2 inner
380.3.j.a.227.77 yes 232 4.3 odd 2 inner
380.3.j.a.227.98 yes 232 19.18 odd 2 inner
380.3.j.a.303.19 yes 232 380.303 odd 4 inner
380.3.j.a.303.40 yes 232 5.3 odd 4 inner
380.3.j.a.303.77 yes 232 95.18 even 4 inner
380.3.j.a.303.98 yes 232 20.3 even 4 inner