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Properties

Label 380.3.g.c.189.10
Level $380$
Weight $3$
Character 380.189
Analytic conductor $10.354$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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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(189,380)] 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("380.189"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.3542500457\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 20x^{10} + 44x^{8} - 270x^{6} + 36676x^{4} - 71664x^{2} + 687241 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{19} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 189.10
Root \(-1.63363 - 1.34185i\) of defining polynomial
Character \(\chi\) \(=\) 380.189
Dual form 380.3.g.c.189.9

$q$-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)
 
\(f(q)\) \(=\) \(q+3.26725 q^{3} +(4.26535 + 2.60898i) q^{5} -6.30368i q^{7} +1.67495 q^{9} +4.53070 q^{11} +11.5357 q^{13} +(13.9360 + 8.52420i) q^{15} +1.08572i q^{17} +(16.8558 - 8.76832i) q^{19} -20.5957i q^{21} -31.5184i q^{23} +(11.3865 + 22.2564i) q^{25} -23.9328 q^{27} +38.1324i q^{29} +58.2398i q^{31} +14.8030 q^{33} +(16.4462 - 26.8874i) q^{35} -40.2691 q^{37} +37.6901 q^{39} -18.0249i q^{41} -2.74764i q^{43} +(7.14424 + 4.36990i) q^{45} -76.1195i q^{47} +9.26364 q^{49} +3.54733i q^{51} +8.06781 q^{53} +(19.3251 + 11.8205i) q^{55} +(55.0720 - 28.6483i) q^{57} +95.3957i q^{59} -4.28507 q^{61} -10.5583i q^{63} +(49.2039 + 30.0964i) q^{65} -70.7364 q^{67} -102.979i q^{69} -93.3131i q^{71} -14.2908i q^{73} +(37.2025 + 72.7174i) q^{75} -28.5601i q^{77} -6.11815i q^{79} -93.2691 q^{81} +89.0257i q^{83} +(-2.83262 + 4.63098i) q^{85} +124.588i q^{87} -92.3366i q^{89} -72.7174i q^{91} +190.284i q^{93} +(94.7721 + 6.57633i) q^{95} -64.6843 q^{97} +7.58869 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{5} + 76 q^{9} - 24 q^{11} + 68 q^{19} - 76 q^{25} - 32 q^{35} - 264 q^{39} + 220 q^{45} + 212 q^{49} + 176 q^{55} - 600 q^{61} + 492 q^{81} + 408 q^{85} + 124 q^{95} + 136 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.26725 1.08908 0.544542 − 0.838733i \(-0.316703\pi\)
0.544542 + 0.838733i \(0.316703\pi\)
\(4\) 0 0
\(5\) 4.26535 + 2.60898i 0.853070 + 0.521796i
\(6\) 0 0
\(7\) − 6.30368i − 0.900525i −0.892896 − 0.450263i \(-0.851330\pi\)
0.892896 − 0.450263i \(-0.148670\pi\)
\(8\) 0 0
\(9\) 1.67495 0.186105
\(10\) 0 0
\(11\) 4.53070 0.411882 0.205941 − 0.978564i \(-0.433974\pi\)
0.205941 + 0.978564i \(0.433974\pi\)
\(12\) 0 0
\(13\) 11.5357 0.887362 0.443681 − 0.896185i \(-0.353672\pi\)
0.443681 + 0.896185i \(0.353672\pi\)
\(14\) 0 0
\(15\) 13.9360 + 8.52420i 0.929066 + 0.568280i
\(16\) 0 0
\(17\) 1.08572i 0.0638659i 0.999490 + 0.0319330i \(0.0101663\pi\)
−0.999490 + 0.0319330i \(0.989834\pi\)
\(18\) 0 0
\(19\) 16.8558 − 8.76832i 0.887145 − 0.461491i
\(20\) 0 0
\(21\) − 20.5957i − 0.980748i
\(22\) 0 0
\(23\) − 31.5184i − 1.37036i −0.728372 − 0.685182i \(-0.759722\pi\)
0.728372 − 0.685182i \(-0.240278\pi\)
\(24\) 0 0
\(25\) 11.3865 + 22.2564i 0.455458 + 0.890257i
\(26\) 0 0
\(27\) −23.9328 −0.886400
\(28\) 0 0
\(29\) 38.1324i 1.31491i 0.753494 + 0.657454i \(0.228367\pi\)
−0.753494 + 0.657454i \(0.771633\pi\)
\(30\) 0 0
\(31\) 58.2398i 1.87870i 0.342955 + 0.939352i \(0.388572\pi\)
−0.342955 + 0.939352i \(0.611428\pi\)
\(32\) 0 0
\(33\) 14.8030 0.448575
\(34\) 0 0
\(35\) 16.4462 − 26.8874i 0.469890 − 0.768212i
\(36\) 0 0
\(37\) −40.2691 −1.08835 −0.544177 − 0.838971i \(-0.683158\pi\)
−0.544177 + 0.838971i \(0.683158\pi\)
\(38\) 0 0
\(39\) 37.6901 0.966413
\(40\) 0 0
\(41\) − 18.0249i − 0.439632i −0.975541 − 0.219816i \(-0.929454\pi\)
0.975541 − 0.219816i \(-0.0705456\pi\)
\(42\) 0 0
\(43\) − 2.74764i − 0.0638986i −0.999489 − 0.0319493i \(-0.989828\pi\)
0.999489 − 0.0319493i \(-0.0101715\pi\)
\(44\) 0 0
\(45\) 7.14424 + 4.36990i 0.158761 + 0.0971090i
\(46\) 0 0
\(47\) − 76.1195i − 1.61956i −0.586731 − 0.809782i \(-0.699586\pi\)
0.586731 − 0.809782i \(-0.300414\pi\)
\(48\) 0 0
\(49\) 9.26364 0.189054
\(50\) 0 0
\(51\) 3.54733i 0.0695554i
\(52\) 0 0
\(53\) 8.06781 0.152223 0.0761115 − 0.997099i \(-0.475750\pi\)
0.0761115 + 0.997099i \(0.475750\pi\)
\(54\) 0 0
\(55\) 19.3251 + 11.8205i 0.351365 + 0.214918i
\(56\) 0 0
\(57\) 55.0720 − 28.6483i 0.966176 − 0.502602i
\(58\) 0 0
\(59\) 95.3957i 1.61688i 0.588581 + 0.808438i \(0.299687\pi\)
−0.588581 + 0.808438i \(0.700313\pi\)
\(60\) 0 0
\(61\) −4.28507 −0.0702470 −0.0351235 − 0.999383i \(-0.511182\pi\)
−0.0351235 + 0.999383i \(0.511182\pi\)
\(62\) 0 0
\(63\) − 10.5583i − 0.167593i
\(64\) 0 0
\(65\) 49.2039 + 30.0964i 0.756982 + 0.463022i
\(66\) 0 0
\(67\) −70.7364 −1.05577 −0.527883 − 0.849317i \(-0.677014\pi\)
−0.527883 + 0.849317i \(0.677014\pi\)
\(68\) 0 0
\(69\) − 102.979i − 1.49244i
\(70\) 0 0
\(71\) − 93.3131i − 1.31427i −0.753773 − 0.657135i \(-0.771768\pi\)
0.753773 − 0.657135i \(-0.228232\pi\)
\(72\) 0 0
\(73\) − 14.2908i − 0.195765i −0.995198 − 0.0978824i \(-0.968793\pi\)
0.995198 − 0.0978824i \(-0.0312069\pi\)
\(74\) 0 0
\(75\) 37.2025 + 72.7174i 0.496033 + 0.969565i
\(76\) 0 0
\(77\) − 28.5601i − 0.370910i
\(78\) 0 0
\(79\) − 6.11815i − 0.0774449i −0.999250 − 0.0387224i \(-0.987671\pi\)
0.999250 − 0.0387224i \(-0.0123288\pi\)
\(80\) 0 0
\(81\) −93.2691 −1.15147
\(82\) 0 0
\(83\) 89.0257i 1.07260i 0.844028 + 0.536299i \(0.180178\pi\)
−0.844028 + 0.536299i \(0.819822\pi\)
\(84\) 0 0
\(85\) −2.83262 + 4.63098i −0.0333250 + 0.0544821i
\(86\) 0 0
\(87\) 124.588i 1.43205i
\(88\) 0 0
\(89\) − 92.3366i − 1.03749i −0.854929 − 0.518745i \(-0.826399\pi\)
0.854929 − 0.518745i \(-0.173601\pi\)
\(90\) 0 0
\(91\) − 72.7174i − 0.799092i
\(92\) 0 0
\(93\) 190.284i 2.04607i
\(94\) 0 0
\(95\) 94.7721 + 6.57633i 0.997601 + 0.0692245i
\(96\) 0 0
\(97\) −64.6843 −0.666849 −0.333424 − 0.942777i \(-0.608204\pi\)
−0.333424 + 0.942777i \(0.608204\pi\)
\(98\) 0 0
\(99\) 7.58869 0.0766535
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.g.c.189.10 yes 12
3.2 odd 2 3420.3.h.e.2089.2 12
5.2 odd 4 1900.3.e.e.1101.4 12
5.3 odd 4 1900.3.e.e.1101.9 12
5.4 even 2 inner 380.3.g.c.189.3 ✓ 12
15.14 odd 2 3420.3.h.e.2089.3 12
19.18 odd 2 inner 380.3.g.c.189.4 yes 12
57.56 even 2 3420.3.h.e.2089.1 12
95.18 even 4 1900.3.e.e.1101.3 12
95.37 even 4 1900.3.e.e.1101.10 12
95.94 odd 2 inner 380.3.g.c.189.9 yes 12
285.284 even 2 3420.3.h.e.2089.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.g.c.189.3 ✓ 12 5.4 even 2 inner
380.3.g.c.189.4 yes 12 19.18 odd 2 inner
380.3.g.c.189.9 yes 12 95.94 odd 2 inner
380.3.g.c.189.10 yes 12 1.1 even 1 trivial
1900.3.e.e.1101.3 12 95.18 even 4
1900.3.e.e.1101.4 12 5.2 odd 4
1900.3.e.e.1101.9 12 5.3 odd 4
1900.3.e.e.1101.10 12 95.37 even 4
3420.3.h.e.2089.1 12 57.56 even 2
3420.3.h.e.2089.2 12 3.2 odd 2
3420.3.h.e.2089.3 12 15.14 odd 2
3420.3.h.e.2089.4 12 285.284 even 2