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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(159,380)] 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("380.159"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 2])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.p (of order \(6\), 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 \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 239.11
Character \(\chi\) \(=\) 380.239
Dual form 380.3.p.a.159.11

$q$-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+(-1.92793 - 0.532049i) q^{2} +(-1.00965 + 1.74877i) q^{3} +(3.43385 + 2.05151i) q^{4} +(-4.76464 - 1.51598i) q^{5} +(2.87698 - 2.83433i) q^{6} +3.07442 q^{7} +(-5.52873 - 5.78214i) q^{8} +(2.46120 + 4.26292i) q^{9} +(8.37933 + 5.45773i) q^{10} -3.77070i q^{11} +(-7.05462 + 3.93371i) q^{12} +(-9.11726 + 5.26385i) q^{13} +(-5.92727 - 1.63574i) q^{14} +(7.46174 - 6.80166i) q^{15} +(7.58263 + 14.0891i) q^{16} +(9.93069 + 5.73349i) q^{17} +(-2.47694 - 9.52810i) q^{18} +(-16.4485 + 9.51032i) q^{19} +(-13.2510 - 14.9803i) q^{20} +(-3.10410 + 5.37645i) q^{21} +(-2.00620 + 7.26966i) q^{22} +(-6.96607 - 12.0656i) q^{23} +(15.6937 - 3.83052i) q^{24} +(20.4036 + 14.4462i) q^{25} +(20.3781 - 5.29753i) q^{26} -28.1136 q^{27} +(10.5571 + 6.30719i) q^{28} +(-9.09921 - 15.7603i) q^{29} +(-18.0045 + 9.14313i) q^{30} -50.1003i q^{31} +(-7.12270 - 31.1972i) q^{32} +(6.59410 + 3.80710i) q^{33} +(-16.0952 - 16.3374i) q^{34} +(-14.6485 - 4.66075i) q^{35} +(-0.294031 + 19.6874i) q^{36} +17.0262i q^{37} +(36.7716 - 9.58385i) q^{38} -21.2587i q^{39} +(17.5768 + 35.9313i) q^{40} +(35.7872 - 61.9853i) q^{41} +(8.84502 - 8.71391i) q^{42} +(-2.56904 + 4.44971i) q^{43} +(7.73562 - 12.9480i) q^{44} +(-5.26423 - 24.0424i) q^{45} +(7.01063 + 26.9679i) q^{46} +(-27.8127 - 48.1730i) q^{47} +(-32.2945 - 0.964853i) q^{48} -39.5480 q^{49} +(-31.6507 - 38.7070i) q^{50} +(-20.0531 + 11.5777i) q^{51} +(-42.1061 - 0.628855i) q^{52} +(21.0636 - 12.1611i) q^{53} +(54.2011 + 14.9578i) q^{54} +(-5.71630 + 17.9660i) q^{55} +(-16.9976 - 17.7767i) q^{56} +(-0.0240704 - 38.3668i) q^{57} +(9.15743 + 35.2260i) q^{58} +(-79.5696 - 45.9396i) q^{59} +(39.5761 - 8.04805i) q^{60} +(-2.51799 - 4.36129i) q^{61} +(-26.6558 + 96.5899i) q^{62} +(7.56675 + 13.1060i) q^{63} +(-2.86635 + 63.9358i) q^{64} +(51.4204 - 11.2588i) q^{65} +(-10.6874 - 10.8482i) q^{66} +(-16.9406 - 29.3419i) q^{67} +(22.3382 + 40.0608i) q^{68} +28.1333 q^{69} +(25.7616 + 16.7793i) q^{70} +(-112.787 - 65.1173i) q^{71} +(11.0415 - 37.7995i) q^{72} +(-69.2413 - 39.9765i) q^{73} +(9.05877 - 32.8254i) q^{74} +(-45.8637 + 21.0956i) q^{75} +(-75.9922 - 1.08726i) q^{76} -11.5927i q^{77} +(-11.3107 + 40.9853i) q^{78} +(-6.25869 - 3.61346i) q^{79} +(-14.7697 - 78.6248i) q^{80} +(6.23423 - 10.7980i) q^{81} +(-101.975 + 100.463i) q^{82} +61.2186 q^{83} +(-21.6888 + 12.0939i) q^{84} +(-38.6243 - 42.3727i) q^{85} +(7.32040 - 7.21189i) q^{86} +36.7482 q^{87} +(-21.8027 + 20.8472i) q^{88} +(14.2293 + 24.6458i) q^{89} +(-2.64265 + 49.1530i) q^{90} +(-28.0303 + 16.1833i) q^{91} +(0.832213 - 55.7223i) q^{92} +(87.6139 + 50.5839i) q^{93} +(27.9906 + 107.672i) q^{94} +(92.7887 - 20.3777i) q^{95} +(61.7483 + 19.0424i) q^{96} +(19.9853 + 11.5385i) q^{97} +(76.2458 + 21.0414i) q^{98} +(16.0742 - 9.28045i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 232 q - 2 q^{5} + 8 q^{6} - 328 q^{9} + 20 q^{14} + 12 q^{16} + 92 q^{20} - 40 q^{21} - 134 q^{24} - 2 q^{25} + 28 q^{26} - 4 q^{29} + 268 q^{30} - 70 q^{34} + 12 q^{36} - 42 q^{40} - 12 q^{41} + 98 q^{44}+ \cdots - 628 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.92793 − 0.532049i −0.963966 − 0.266024i
\(3\) −1.00965 + 1.74877i −0.336551 + 0.582924i −0.983782 − 0.179371i \(-0.942594\pi\)
0.647230 + 0.762295i \(0.275927\pi\)
\(4\) 3.43385 + 2.05151i 0.858462 + 0.512877i
\(5\) −4.76464 − 1.51598i −0.952928 − 0.303196i
\(6\) 2.87698 − 2.83433i 0.479496 − 0.472388i
\(7\) 3.07442 0.439202 0.219601 − 0.975590i \(-0.429524\pi\)
0.219601 + 0.975590i \(0.429524\pi\)
\(8\) −5.52873 − 5.78214i −0.691091 − 0.722768i
\(9\) 2.46120 + 4.26292i 0.273466 + 0.473658i
\(10\) 8.37933 + 5.45773i 0.837933 + 0.545773i
\(11\) − 3.77070i − 0.342791i −0.985202 − 0.171396i \(-0.945172\pi\)
0.985202 − 0.171396i \(-0.0548276\pi\)
\(12\) −7.05462 + 3.93371i −0.587885 + 0.327809i
\(13\) −9.11726 + 5.26385i −0.701328 + 0.404912i −0.807842 − 0.589399i \(-0.799364\pi\)
0.106514 + 0.994311i \(0.466031\pi\)
\(14\) −5.92727 − 1.63574i −0.423376 − 0.116839i
\(15\) 7.46174 − 6.80166i 0.497449 − 0.453444i
\(16\) 7.58263 + 14.0891i 0.473915 + 0.880571i
\(17\) 9.93069 + 5.73349i 0.584158 + 0.337264i 0.762784 − 0.646653i \(-0.223832\pi\)
−0.178626 + 0.983917i \(0.557165\pi\)
\(18\) −2.47694 − 9.52810i −0.137608 − 0.529339i
\(19\) −16.4485 + 9.51032i −0.865712 + 0.500543i
\(20\) −13.2510 − 14.9803i −0.662551 − 0.749017i
\(21\) −3.10410 + 5.37645i −0.147814 + 0.256022i
\(22\) −2.00620 + 7.26966i −0.0911908 + 0.330439i
\(23\) −6.96607 − 12.0656i −0.302872 − 0.524591i 0.673913 − 0.738811i \(-0.264612\pi\)
−0.976785 + 0.214220i \(0.931279\pi\)
\(24\) 15.6937 − 3.83052i 0.653906 − 0.159605i
\(25\) 20.4036 + 14.4462i 0.816145 + 0.577848i
\(26\) 20.3781 − 5.29753i 0.783773 − 0.203751i
\(27\) −28.1136 −1.04124
\(28\) 10.5571 + 6.30719i 0.377039 + 0.225257i
\(29\) −9.09921 − 15.7603i −0.313766 − 0.543459i 0.665408 − 0.746480i \(-0.268257\pi\)
−0.979174 + 0.203021i \(0.934924\pi\)
\(30\) −18.0045 + 9.14313i −0.600151 + 0.304771i
\(31\) − 50.1003i − 1.61614i −0.589088 − 0.808069i \(-0.700513\pi\)
0.589088 − 0.808069i \(-0.299487\pi\)
\(32\) −7.12270 − 31.1972i −0.222584 − 0.974913i
\(33\) 6.59410 + 3.80710i 0.199821 + 0.115367i
\(34\) −16.0952 − 16.3374i −0.473389 − 0.480511i
\(35\) −14.6485 − 4.66075i −0.418528 − 0.133164i
\(36\) −0.294031 + 19.6874i −0.00816753 + 0.546872i
\(37\) 17.0262i 0.460168i 0.973171 + 0.230084i \(0.0739000\pi\)
−0.973171 + 0.230084i \(0.926100\pi\)
\(38\) 36.7716 − 9.58385i 0.967673 − 0.252207i
\(39\) − 21.2587i − 0.545094i
\(40\) 17.5768 + 35.9313i 0.439420 + 0.898282i
\(41\) 35.7872 − 61.9853i 0.872859 − 1.51184i 0.0138327 − 0.999904i \(-0.495597\pi\)
0.859026 − 0.511932i \(-0.171070\pi\)
\(42\) 8.84502 − 8.71391i 0.210596 − 0.207474i
\(43\) −2.56904 + 4.44971i −0.0597452 + 0.103482i −0.894351 − 0.447366i \(-0.852362\pi\)
0.834606 + 0.550848i \(0.185695\pi\)
\(44\) 7.73562 − 12.9480i 0.175810 − 0.294273i
\(45\) −5.26423 − 24.0424i −0.116983 − 0.534276i
\(46\) 7.01063 + 26.9679i 0.152405 + 0.586259i
\(47\) −27.8127 − 48.1730i −0.591760 − 1.02496i −0.993995 − 0.109422i \(-0.965100\pi\)
0.402236 − 0.915536i \(-0.368233\pi\)
\(48\) −32.2945 − 0.964853i −0.672802 − 0.0201011i
\(49\) −39.5480 −0.807101
\(50\) −31.6507 − 38.7070i −0.633015 − 0.774140i
\(51\) −20.0531 + 11.5777i −0.393198 + 0.227013i
\(52\) −42.1061 − 0.628855i −0.809733 − 0.0120934i
\(53\) 21.0636 − 12.1611i 0.397427 − 0.229455i −0.287946 − 0.957647i \(-0.592972\pi\)
0.685373 + 0.728192i \(0.259639\pi\)
\(54\) 54.2011 + 14.9578i 1.00372 + 0.276996i
\(55\) −5.71630 + 17.9660i −0.103933 + 0.326655i
\(56\) −16.9976 − 17.7767i −0.303529 − 0.317441i
\(57\) −0.0240704 − 38.3668i −0.000422287 − 0.673102i
\(58\) 9.15743 + 35.2260i 0.157887 + 0.607345i
\(59\) −79.5696 − 45.9396i −1.34864 − 0.778636i −0.360581 − 0.932728i \(-0.617422\pi\)
−0.988057 + 0.154091i \(0.950755\pi\)
\(60\) 39.5761 − 8.04805i 0.659602 − 0.134134i
\(61\) −2.51799 − 4.36129i −0.0412786 − 0.0714966i 0.844648 − 0.535322i \(-0.179810\pi\)
−0.885927 + 0.463826i \(0.846476\pi\)
\(62\) −26.6558 + 96.5899i −0.429932 + 1.55790i
\(63\) 7.56675 + 13.1060i 0.120107 + 0.208032i
\(64\) −2.86635 + 63.9358i −0.0447867 + 0.998997i
\(65\) 51.4204 − 11.2588i 0.791083 − 0.173212i
\(66\) −10.6874 − 10.8482i −0.161931 − 0.164367i
\(67\) −16.9406 − 29.3419i −0.252844 − 0.437939i 0.711464 − 0.702723i \(-0.248033\pi\)
−0.964308 + 0.264784i \(0.914699\pi\)
\(68\) 22.3382 + 40.0608i 0.328503 + 0.589130i
\(69\) 28.1333 0.407728
\(70\) 25.7616 + 16.7793i 0.368022 + 0.239705i
\(71\) −112.787 − 65.1173i −1.58854 − 0.917146i −0.993548 − 0.113417i \(-0.963820\pi\)
−0.594996 − 0.803729i \(-0.702846\pi\)
\(72\) 11.0415 − 37.7995i 0.153354 − 0.524993i
\(73\) −69.2413 − 39.9765i −0.948512 − 0.547623i −0.0558933 − 0.998437i \(-0.517801\pi\)
−0.892618 + 0.450813i \(0.851134\pi\)
\(74\) 9.05877 − 32.8254i 0.122416 − 0.443586i
\(75\) −45.8637 + 21.0956i −0.611516 + 0.281275i
\(76\) −75.9922 − 1.08726i −0.999898 − 0.0143060i
\(77\) − 11.5927i − 0.150555i
\(78\) −11.3107 + 40.9853i −0.145008 + 0.525453i
\(79\) −6.25869 − 3.61346i −0.0792239 − 0.0457400i 0.459865 − 0.887989i \(-0.347898\pi\)
−0.539089 + 0.842249i \(0.681231\pi\)
\(80\) −14.7697 − 78.6248i −0.184621 − 0.982810i
\(81\) 6.23423 − 10.7980i 0.0769658 − 0.133309i
\(82\) −101.975 + 100.463i −1.24359 + 1.22516i
\(83\) 61.2186 0.737574 0.368787 − 0.929514i \(-0.379773\pi\)
0.368787 + 0.929514i \(0.379773\pi\)
\(84\) −21.6888 + 12.0939i −0.258200 + 0.143974i
\(85\) −38.6243 − 42.3727i −0.454404 − 0.498503i
\(86\) 7.32040 − 7.21189i 0.0851210 − 0.0838592i
\(87\) 36.7482 0.422393
\(88\) −21.8027 + 20.8472i −0.247758 + 0.236900i
\(89\) 14.2293 + 24.6458i 0.159879 + 0.276919i 0.934825 − 0.355109i \(-0.115556\pi\)
−0.774946 + 0.632028i \(0.782223\pi\)
\(90\) −2.64265 + 49.1530i −0.0293627 + 0.546144i
\(91\) −28.0303 + 16.1833i −0.308025 + 0.177838i
\(92\) 0.832213 − 55.7223i 0.00904579 − 0.605677i
\(93\) 87.6139 + 50.5839i 0.942085 + 0.543913i
\(94\) 27.9906 + 107.672i 0.297773 + 1.14545i
\(95\) 92.7887 − 20.3777i 0.976724 − 0.214502i
\(96\) 61.7483 + 19.0424i 0.643211 + 0.198359i
\(97\) 19.9853 + 11.5385i 0.206034 + 0.118954i 0.599467 − 0.800399i \(-0.295379\pi\)
−0.393433 + 0.919353i \(0.628713\pi\)
\(98\) 76.2458 + 21.0414i 0.778018 + 0.214709i
\(99\) 16.0742 − 9.28045i 0.162366 − 0.0937419i
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.p.a.239.11 yes 232
4.3 odd 2 inner 380.3.p.a.239.30 yes 232
5.4 even 2 inner 380.3.p.a.239.106 yes 232
19.7 even 3 inner 380.3.p.a.159.87 yes 232
20.19 odd 2 inner 380.3.p.a.239.87 yes 232
76.7 odd 6 inner 380.3.p.a.159.106 yes 232
95.64 even 6 inner 380.3.p.a.159.30 yes 232
380.159 odd 6 inner 380.3.p.a.159.11 ✓ 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.11 ✓ 232 380.159 odd 6 inner
380.3.p.a.159.30 yes 232 95.64 even 6 inner
380.3.p.a.159.87 yes 232 19.7 even 3 inner
380.3.p.a.159.106 yes 232 76.7 odd 6 inner
380.3.p.a.239.11 yes 232 1.1 even 1 trivial
380.3.p.a.239.30 yes 232 4.3 odd 2 inner
380.3.p.a.239.87 yes 232 20.19 odd 2 inner
380.3.p.a.239.106 yes 232 5.4 even 2 inner