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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 239.106
Character \(\chi\) \(=\) 380.239
Dual form 380.3.p.a.159.106

$q$-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} +(3.69520 + 3.36831i) q^{5} +(2.87698 - 2.83433i) q^{6} -3.07442 q^{7} +(5.52873 + 5.78214i) q^{8} +(2.46120 + 4.26292i) q^{9} +(5.33199 + 8.45990i) q^{10} -3.77070i q^{11} +(7.05462 - 3.93371i) q^{12} +(9.11726 - 5.26385i) q^{13} +(-5.92727 - 1.63574i) q^{14} +(9.62128 - 3.06123i) 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} +(5.77863 + 19.1470i) 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} +(2.30896 + 24.8931i) q^{25} +(20.3781 - 5.29753i) q^{26} +28.1136 q^{27} +(-10.5571 - 6.30719i) q^{28} +(-9.09921 - 15.7603i) q^{29} +(20.1779 - 0.782853i) q^{30} -50.1003i q^{31} +(7.12270 + 31.1972i) q^{32} +(-6.59410 - 3.80710i) q^{33} +(-16.0952 - 16.3374i) q^{34} +(-11.3606 - 10.3556i) q^{35} +(-0.294031 + 19.6874i) q^{36} -17.0262i q^{37} +(-36.7716 + 9.58385i) q^{38} -21.2587i q^{39} +(0.953678 + 39.9886i) 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} +(-8.79285 + 49.2208i) 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} +(12.7009 - 13.9335i) 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} +(39.3181 + 9.22633i) 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} +(-16.3927 - 26.0093i) 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} +(-19.4373 + 77.6028i) q^{80} +(6.23423 - 10.7980i) q^{81} +(101.975 - 100.463i) q^{82} -61.2186 q^{83} +(-21.6888 + 12.0939i) q^{84} +(-17.3837 - 54.6360i) q^{85} +(7.32040 - 7.21189i) q^{86} -36.7482 q^{87} +(21.8027 - 20.8472i) q^{88} +(14.2293 + 24.6458i) q^{89} +(-22.9408 + 43.5513i) 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} +(-92.8142 - 20.2612i) 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 −0.647230 − 0.762295i \(-0.724073\pi\)
0.983782 + 0.179371i \(0.0574061\pi\)
\(4\) 3.43385 + 2.05151i 0.858462 + 0.512877i
\(5\) 3.69520 + 3.36831i 0.739039 + 0.673662i
\(6\) 2.87698 − 2.83433i 0.479496 − 0.472388i
\(7\) −3.07442 −0.439202 −0.219601 − 0.975590i \(-0.570476\pi\)
−0.219601 + 0.975590i \(0.570476\pi\)
\(8\) 5.52873 + 5.78214i 0.691091 + 0.722768i
\(9\) 2.46120 + 4.26292i 0.273466 + 0.473658i
\(10\) 5.33199 + 8.45990i 0.533199 + 0.845990i
\(11\) − 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 −0.106514 − 0.994311i \(-0.533969\pi\)
0.807842 + 0.589399i \(0.200636\pi\)
\(14\) −5.92727 − 1.63574i −0.423376 − 0.116839i
\(15\) 9.62128 − 3.06123i 0.641419 − 0.204082i
\(16\) 7.58263 + 14.0891i 0.473915 + 0.880571i
\(17\) −9.93069 − 5.73349i −0.584158 − 0.337264i 0.178626 − 0.983917i \(-0.442835\pi\)
−0.762784 + 0.646653i \(0.776168\pi\)
\(18\) 2.47694 + 9.52810i 0.137608 + 0.529339i
\(19\) −16.4485 + 9.51032i −0.865712 + 0.500543i
\(20\) 5.77863 + 19.1470i 0.288931 + 0.957350i
\(21\) −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 0.976785 − 0.214220i \(-0.0687210\pi\)
−0.673913 + 0.738811i \(0.735388\pi\)
\(24\) 15.6937 − 3.83052i 0.653906 − 0.159605i
\(25\) 2.30896 + 24.8931i 0.0923583 + 0.995726i
\(26\) 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 \(0.934924\pi\)
\(30\) 20.1779 − 0.782853i 0.672597 − 0.0260951i
\(31\) − 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
\(35\) −11.3606 − 10.3556i −0.324588 − 0.295874i
\(36\) −0.294031 + 19.6874i −0.00816753 + 0.546872i
\(37\) − 17.0262i − 0.460168i −0.973171 − 0.230084i \(-0.926100\pi\)
0.973171 − 0.230084i \(-0.0739000\pi\)
\(38\) −36.7716 + 9.58385i −0.967673 + 0.252207i
\(39\) − 21.2587i − 0.545094i
\(40\) 0.953678 + 39.9886i 0.0238419 + 0.999716i
\(41\) 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 −0.834606 − 0.550848i \(-0.814305\pi\)
0.894351 + 0.447366i \(0.147638\pi\)
\(44\) 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 \(0.0348999\pi\)
−0.402236 + 0.915536i \(0.631767\pi\)
\(48\) 32.2945 + 0.964853i 0.672802 + 0.0201011i
\(49\) −39.5480 −0.807101
\(50\) −8.79285 + 49.2208i −0.175857 + 0.984416i
\(51\) −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 −0.685373 − 0.728192i \(-0.740361\pi\)
0.287946 + 0.957647i \(0.407028\pi\)
\(54\) 54.2011 + 14.9578i 1.00372 + 0.276996i
\(55\) 12.7009 − 13.9335i 0.230925 − 0.253336i
\(56\) −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 \(0.950755\pi\)
\(60\) 39.3181 + 9.22633i 0.655302 + 0.153772i
\(61\) −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 0.964308 − 0.264784i \(-0.0853007\pi\)
−0.711464 + 0.702723i \(0.751967\pi\)
\(68\) −22.3382 − 40.0608i −0.328503 − 0.589130i
\(69\) 28.1333 0.407728
\(70\) −16.3927 − 26.0093i −0.234182 − 0.371561i
\(71\) −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 0.892618 − 0.450813i \(-0.148866\pi\)
0.0558933 + 0.998437i \(0.482199\pi\)
\(74\) 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 \(0.681231\pi\)
\(80\) −19.4373 + 77.6028i −0.242966 + 0.970035i
\(81\) 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 \(-0.620227\pi\)
−0.368787 + 0.929514i \(0.620227\pi\)
\(84\) −21.6888 + 12.0939i −0.258200 + 0.143974i
\(85\) −17.3837 − 54.6360i −0.204514 − 0.642777i
\(86\) 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 \(0.782223\pi\)
\(90\) −22.9408 + 43.5513i −0.254898 + 0.483904i
\(91\) −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
\(95\) −92.8142 − 20.2612i −0.976992 − 0.213276i
\(96\) 61.7483 + 19.0424i 0.643211 + 0.198359i
\(97\) −19.9853 − 11.5385i −0.206034 − 0.118954i 0.393433 − 0.919353i \(-0.371287\pi\)
−0.599467 + 0.800399i \(0.704621\pi\)
\(98\) −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.239.106 yes 232
4.3 odd 2 inner 380.3.p.a.239.87 yes 232
5.4 even 2 inner 380.3.p.a.239.11 yes 232
19.7 even 3 inner 380.3.p.a.159.30 yes 232
20.19 odd 2 inner 380.3.p.a.239.30 yes 232
76.7 odd 6 inner 380.3.p.a.159.11 ✓ 232
95.64 even 6 inner 380.3.p.a.159.87 yes 232
380.159 odd 6 inner 380.3.p.a.159.106 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.11 ✓ 232 76.7 odd 6 inner
380.3.p.a.159.30 yes 232 19.7 even 3 inner
380.3.p.a.159.87 yes 232 95.64 even 6 inner
380.3.p.a.159.106 yes 232 380.159 odd 6 inner
380.3.p.a.239.11 yes 232 5.4 even 2 inner
380.3.p.a.239.30 yes 232 20.19 odd 2 inner
380.3.p.a.239.87 yes 232 4.3 odd 2 inner
380.3.p.a.239.106 yes 232 1.1 even 1 trivial