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Properties

Label 380.3.p.a.159.11
Level $380$
Weight $3$
Character 380.159
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(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 159.11
Character \(\chi\) \(=\) 380.159
Dual form 380.3.p.a.239.11

$q$-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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

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

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 \(-0.275927\pi\)
−0.983782 + 0.179371i \(0.942594\pi\)
\(4\) 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 \(0.0548276\pi\)
−0.985202 + 0.171396i \(0.945172\pi\)
\(12\) −7.05462 − 3.93371i −0.587885 − 0.327809i
\(13\) −9.11726 − 5.26385i −0.701328 − 0.404912i 0.106514 − 0.994311i \(-0.466031\pi\)
−0.807842 + 0.589399i \(0.799364\pi\)
\(14\) −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 −0.178626 − 0.983917i \(-0.557165\pi\)
0.762784 + 0.646653i \(0.223832\pi\)
\(18\) −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 −0.976785 − 0.214220i \(-0.931279\pi\)
0.673913 + 0.738811i \(0.264612\pi\)
\(24\) 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 −0.979174 − 0.203021i \(-0.934924\pi\)
0.665408 + 0.746480i \(0.268257\pi\)
\(30\) −18.0045 − 9.14313i −0.600151 − 0.304771i
\(31\) 50.1003i 1.61614i 0.589088 + 0.808069i \(0.299487\pi\)
−0.589088 + 0.808069i \(0.700513\pi\)
\(32\) −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 \(-0.926100\pi\)
0.973171 − 0.230084i \(-0.0739000\pi\)
\(38\) 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 0.859026 + 0.511932i \(0.171070\pi\)
0.0138327 + 0.999904i \(0.495597\pi\)
\(42\) 8.84502 + 8.71391i 0.210596 + 0.207474i
\(43\) −2.56904 − 4.44971i −0.0597452 − 0.103482i 0.834606 − 0.550848i \(-0.185695\pi\)
−0.894351 + 0.447366i \(0.852362\pi\)
\(44\) 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 0.402236 + 0.915536i \(0.368233\pi\)
−0.993995 + 0.109422i \(0.965100\pi\)
\(48\) −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 0.685373 − 0.728192i \(-0.259639\pi\)
−0.287946 + 0.957647i \(0.592972\pi\)
\(54\) 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 −0.988057 − 0.154091i \(-0.950755\pi\)
−0.360581 + 0.932728i \(0.617422\pi\)
\(60\) 39.5761 + 8.04805i 0.659602 + 0.134134i
\(61\) −2.51799 + 4.36129i −0.0412786 + 0.0714966i −0.885927 − 0.463826i \(-0.846476\pi\)
0.844648 + 0.535322i \(0.179810\pi\)
\(62\) −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 \(-0.914699\pi\)
0.711464 + 0.702723i \(0.248033\pi\)
\(68\) 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 −0.594996 + 0.803729i \(0.702846\pi\)
−0.993548 + 0.113417i \(0.963820\pi\)
\(72\) 11.0415 + 37.7995i 0.153354 + 0.524993i
\(73\) −69.2413 + 39.9765i −0.948512 + 0.547623i −0.892618 − 0.450813i \(-0.851134\pi\)
−0.0558933 + 0.998437i \(0.517801\pi\)
\(74\) 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 −0.539089 − 0.842249i \(-0.681231\pi\)
0.459865 + 0.887989i \(0.347898\pi\)
\(80\) −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 −0.774946 − 0.632028i \(-0.782223\pi\)
0.934825 + 0.355109i \(0.115556\pi\)
\(90\) −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 −0.393433 − 0.919353i \(-0.628713\pi\)
0.599467 + 0.800399i \(0.295379\pi\)
\(98\) 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.159.11 ✓ 232
4.3 odd 2 inner 380.3.p.a.159.30 yes 232
5.4 even 2 inner 380.3.p.a.159.106 yes 232
19.11 even 3 inner 380.3.p.a.239.87 yes 232
20.19 odd 2 inner 380.3.p.a.159.87 yes 232
76.11 odd 6 inner 380.3.p.a.239.106 yes 232
95.49 even 6 inner 380.3.p.a.239.30 yes 232
380.239 odd 6 inner 380.3.p.a.239.11 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.11 ✓ 232 1.1 even 1 trivial
380.3.p.a.159.30 yes 232 4.3 odd 2 inner
380.3.p.a.159.87 yes 232 20.19 odd 2 inner
380.3.p.a.159.106 yes 232 5.4 even 2 inner
380.3.p.a.239.11 yes 232 380.239 odd 6 inner
380.3.p.a.239.30 yes 232 95.49 even 6 inner
380.3.p.a.239.87 yes 232 19.11 even 3 inner
380.3.p.a.239.106 yes 232 76.11 odd 6 inner