Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,2,Mod(179,380)] 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("380.179"); S:= CuspForms(chi, 2); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.s (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: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(56\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 259.8
Character \(\chi\) \(=\) 380.259
Dual form 380.2.s.a.179.8

$q$-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+(-1.31940 + 0.509093i) q^{2} +(0.369609 - 0.213394i) q^{3} +(1.48165 - 1.34340i) q^{4} +(2.18710 + 0.465374i) q^{5} +(-0.379026 + 0.469718i) q^{6} -3.44511 q^{7} +(-1.27098 + 2.52678i) q^{8} +(-1.40893 + 2.44033i) q^{9} +(-3.12259 + 0.499423i) q^{10} +2.55436i q^{11} +(0.260958 - 0.812706i) q^{12} +(-3.08972 + 5.35155i) q^{13} +(4.54549 - 1.75388i) q^{14} +(0.907681 - 0.294708i) q^{15} +(0.390567 - 3.98089i) q^{16} +(4.70397 - 2.71584i) q^{17} +(0.616586 - 3.93706i) q^{18} +(4.12479 + 1.40931i) q^{19} +(3.86570 - 2.24863i) q^{20} +(-1.27334 + 0.735166i) q^{21} +(-1.30041 - 3.37023i) q^{22} +(-1.75092 + 3.03269i) q^{23} +(0.0694343 + 1.20514i) q^{24} +(4.56685 + 2.03564i) q^{25} +(1.35215 - 8.63380i) q^{26} +2.48299i q^{27} +(-5.10445 + 4.62816i) q^{28} +(-3.80617 - 2.19750i) q^{29} +(-1.04756 + 0.850933i) q^{30} +3.36388 q^{31} +(1.51133 + 5.45123i) q^{32} +(0.545084 + 0.944113i) q^{33} +(-4.82382 + 5.97804i) q^{34} +(-7.53483 - 1.60327i) q^{35} +(1.19080 + 5.50846i) q^{36} +2.93703 q^{37} +(-6.15972 + 0.240458i) q^{38} +2.63731i q^{39} +(-3.95566 + 4.93485i) q^{40} +(-4.83719 + 2.79275i) q^{41} +(1.30579 - 1.61823i) q^{42} +(2.66813 + 4.62133i) q^{43} +(3.43152 + 3.78466i) q^{44} +(-4.21714 + 4.68158i) q^{45} +(0.766254 - 4.89272i) q^{46} +(5.39759 - 9.34890i) q^{47} +(-0.705139 - 1.55472i) q^{48} +4.86881 q^{49} +(-7.06185 - 0.360884i) q^{50} +(1.15909 - 2.00760i) q^{51} +(2.61138 + 12.0798i) q^{52} +(-1.90332 + 3.29665i) q^{53} +(-1.26407 - 3.27606i) q^{54} +(-1.18873 + 5.58665i) q^{55} +(4.37866 - 8.70504i) q^{56} +(1.82529 - 0.359312i) q^{57} +(6.14061 + 0.961687i) q^{58} +(-6.09481 - 10.5565i) q^{59} +(0.948955 - 1.65603i) q^{60} +(-4.11002 + 7.11877i) q^{61} +(-4.43831 + 1.71253i) q^{62} +(4.85391 - 8.40722i) q^{63} +(-4.76923 - 6.42296i) q^{64} +(-9.24801 + 10.2665i) q^{65} +(-1.19983 - 0.968167i) q^{66} +(0.587816 + 0.339376i) q^{67} +(3.32118 - 10.3432i) q^{68} +1.49455i q^{69} +(10.7577 - 1.72057i) q^{70} +(-3.36699 - 5.83180i) q^{71} +(-4.37547 - 6.66165i) q^{72} +(3.37686 - 1.94963i) q^{73} +(-3.87513 + 1.49522i) q^{74} +(2.12234 - 0.222146i) q^{75} +(8.00474 - 3.45313i) q^{76} -8.80005i q^{77} +(-1.34263 - 3.47967i) q^{78} +(4.09026 + 7.08453i) q^{79} +(2.70681 - 8.52486i) q^{80} +(-3.69692 - 6.40326i) q^{81} +(4.96043 - 6.14735i) q^{82} -1.04506 q^{83} +(-0.899030 + 2.79987i) q^{84} +(11.5520 - 3.75072i) q^{85} +(-5.87302 - 4.73907i) q^{86} -1.87573 q^{87} +(-6.45430 - 3.24653i) q^{88} +(7.94252 + 4.58561i) q^{89} +(3.18074 - 8.32381i) q^{90} +(10.6444 - 18.4367i) q^{91} +(1.47985 + 6.84557i) q^{92} +(1.24332 - 0.717830i) q^{93} +(-2.36214 + 15.0828i) q^{94} +(8.36548 + 5.00187i) q^{95} +(1.72186 + 1.69231i) q^{96} +(-4.94059 - 8.55736i) q^{97} +(-6.42393 + 2.47868i) q^{98} +(-6.23348 - 3.59890i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 112 q - 2 q^{5} - 8 q^{6} + 44 q^{9} + 6 q^{10} - 36 q^{14} - 4 q^{16} + 44 q^{20} - 48 q^{21} + 2 q^{24} - 2 q^{25} - 36 q^{26} - 12 q^{29} - 32 q^{30} - 30 q^{34} + 20 q^{36} - 24 q^{40} - 24 q^{41} - 14 q^{44}+ \cdots - 84 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.31940 + 0.509093i −0.932959 + 0.359983i
\(3\) 0.369609 − 0.213394i 0.213394 − 0.123203i −0.389494 − 0.921029i \(-0.627350\pi\)
0.602888 + 0.797826i \(0.294017\pi\)
\(4\) 1.48165 − 1.34340i 0.740824 − 0.671699i
\(5\) 2.18710 + 0.465374i 0.978103 + 0.208122i
\(6\) −0.379026 + 0.469718i −0.154737 + 0.191761i
\(7\) −3.44511 −1.30213 −0.651065 − 0.759022i \(-0.725678\pi\)
−0.651065 + 0.759022i \(0.725678\pi\)
\(8\) −1.27098 + 2.52678i −0.449359 + 0.893351i
\(9\) −1.40893 + 2.44033i −0.469642 + 0.813444i
\(10\) −3.12259 + 0.499423i −0.987450 + 0.157931i
\(11\) 2.55436i 0.770168i 0.922882 + 0.385084i \(0.125827\pi\)
−0.922882 + 0.385084i \(0.874173\pi\)
\(12\) 0.260958 − 0.812706i 0.0753320 − 0.234608i
\(13\) −3.08972 + 5.35155i −0.856934 + 1.48425i 0.0179060 + 0.999840i \(0.494300\pi\)
−0.874840 + 0.484413i \(0.839033\pi\)
\(14\) 4.54549 − 1.75388i 1.21483 − 0.468745i
\(15\) 0.907681 − 0.294708i 0.234362 − 0.0760933i
\(16\) 0.390567 − 3.98089i 0.0976417 − 0.995222i
\(17\) 4.70397 − 2.71584i 1.14088 − 0.658688i 0.194232 − 0.980956i \(-0.437779\pi\)
0.946648 + 0.322268i \(0.104445\pi\)
\(18\) 0.616586 − 3.93706i 0.145331 − 0.927973i
\(19\) 4.12479 + 1.40931i 0.946291 + 0.323317i
\(20\) 3.86570 − 2.24863i 0.864398 − 0.502809i
\(21\) −1.27334 + 0.735166i −0.277867 + 0.160426i
\(22\) −1.30041 − 3.37023i −0.277247 − 0.718535i
\(23\) −1.75092 + 3.03269i −0.365093 + 0.632360i −0.988791 − 0.149306i \(-0.952296\pi\)
0.623698 + 0.781665i \(0.285629\pi\)
\(24\) 0.0694343 + 1.20514i 0.0141732 + 0.245998i
\(25\) 4.56685 + 2.03564i 0.913371 + 0.407129i
\(26\) 1.35215 − 8.63380i 0.265178 − 1.69323i
\(27\) 2.48299i 0.477851i
\(28\) −5.10445 + 4.62816i −0.964650 + 0.874640i
\(29\) −3.80617 − 2.19750i −0.706789 − 0.408065i 0.103082 − 0.994673i \(-0.467130\pi\)
−0.809871 + 0.586608i \(0.800463\pi\)
\(30\) −1.04756 + 0.850933i −0.191258 + 0.155358i
\(31\) 3.36388 0.604170 0.302085 − 0.953281i \(-0.402317\pi\)
0.302085 + 0.953281i \(0.402317\pi\)
\(32\) 1.51133 + 5.45123i 0.267167 + 0.963650i
\(33\) 0.545084 + 0.944113i 0.0948869 + 0.164349i
\(34\) −4.82382 + 5.97804i −0.827278 + 1.02523i
\(35\) −7.53483 − 1.60327i −1.27362 − 0.271002i
\(36\) 1.19080 + 5.50846i 0.198467 + 0.918077i
\(37\) 2.93703 0.482845 0.241422 − 0.970420i \(-0.422386\pi\)
0.241422 + 0.970420i \(0.422386\pi\)
\(38\) −6.15972 + 0.240458i −0.999239 + 0.0390074i
\(39\) 2.63731i 0.422307i
\(40\) −3.95566 + 4.93485i −0.625445 + 0.780268i
\(41\) −4.83719 + 2.79275i −0.755442 + 0.436155i −0.827657 − 0.561234i \(-0.810327\pi\)
0.0722146 + 0.997389i \(0.476993\pi\)
\(42\) 1.30579 − 1.61823i 0.201487 − 0.249698i
\(43\) 2.66813 + 4.62133i 0.406885 + 0.704746i 0.994539 − 0.104366i \(-0.0332815\pi\)
−0.587653 + 0.809113i \(0.699948\pi\)
\(44\) 3.43152 + 3.78466i 0.517321 + 0.570559i
\(45\) −4.21714 + 4.68158i −0.628654 + 0.697889i
\(46\) 0.766254 − 4.89272i 0.112978 − 0.721393i
\(47\) 5.39759 − 9.34890i 0.787320 − 1.36368i −0.140284 − 0.990111i \(-0.544801\pi\)
0.927603 − 0.373566i \(-0.121865\pi\)
\(48\) −0.705139 − 1.55472i −0.101778 − 0.224404i
\(49\) 4.86881 0.695545
\(50\) −7.06185 − 0.360884i −0.998697 − 0.0510366i
\(51\) 1.15909 − 2.00760i 0.162304 − 0.281120i
\(52\) 2.61138 + 12.0798i 0.362133 + 1.67517i
\(53\) −1.90332 + 3.29665i −0.261441 + 0.452830i −0.966625 − 0.256195i \(-0.917531\pi\)
0.705184 + 0.709024i \(0.250864\pi\)
\(54\) −1.26407 − 3.27606i −0.172018 − 0.445815i
\(55\) −1.18873 + 5.58665i −0.160289 + 0.753303i
\(56\) 4.37866 − 8.70504i 0.585124 − 1.16326i
\(57\) 1.82529 − 0.359312i 0.241766 − 0.0475920i
\(58\) 6.14061 + 0.961687i 0.806301 + 0.126276i
\(59\) −6.09481 − 10.5565i −0.793477 − 1.37434i −0.923802 − 0.382871i \(-0.874935\pi\)
0.130325 − 0.991471i \(-0.458398\pi\)
\(60\) 0.948955 − 1.65603i 0.122510 − 0.213793i
\(61\) −4.11002 + 7.11877i −0.526234 + 0.911465i 0.473299 + 0.880902i \(0.343063\pi\)
−0.999533 + 0.0305625i \(0.990270\pi\)
\(62\) −4.43831 + 1.71253i −0.563666 + 0.217491i
\(63\) 4.85391 − 8.40722i 0.611535 − 1.05921i
\(64\) −4.76923 − 6.42296i −0.596154 − 0.802870i
\(65\) −9.24801 + 10.2665i −1.14707 + 1.27341i
\(66\) −1.19983 − 0.968167i −0.147688 − 0.119173i
\(67\) 0.587816 + 0.339376i 0.0718131 + 0.0414613i 0.535477 − 0.844550i \(-0.320132\pi\)
−0.463664 + 0.886011i \(0.653465\pi\)
\(68\) 3.32118 − 10.3432i 0.402752 − 1.25430i
\(69\) 1.49455i 0.179922i
\(70\) 10.7577 − 1.72057i 1.28579 − 0.205647i
\(71\) −3.36699 − 5.83180i −0.399588 − 0.692107i 0.594087 − 0.804401i \(-0.297514\pi\)
−0.993675 + 0.112294i \(0.964180\pi\)
\(72\) −4.37547 − 6.66165i −0.515654 − 0.785083i
\(73\) 3.37686 − 1.94963i 0.395232 − 0.228187i −0.289193 − 0.957271i \(-0.593387\pi\)
0.684425 + 0.729084i \(0.260053\pi\)
\(74\) −3.87513 + 1.49522i −0.450475 + 0.173816i
\(75\) 2.12234 − 0.222146i 0.245067 − 0.0256512i
\(76\) 8.00474 − 3.45313i 0.918207 − 0.396101i
\(77\) − 8.80005i − 1.00286i
\(78\) −1.34263 − 3.47967i −0.152023 − 0.393995i
\(79\) 4.09026 + 7.08453i 0.460190 + 0.797072i 0.998970 − 0.0453742i \(-0.0144480\pi\)
−0.538780 + 0.842446i \(0.681115\pi\)
\(80\) 2.70681 − 8.52486i 0.302631 − 0.953108i
\(81\) −3.69692 − 6.40326i −0.410769 − 0.711474i
\(82\) 4.96043 − 6.14735i 0.547788 − 0.678861i
\(83\) −1.04506 −0.114710 −0.0573552 − 0.998354i \(-0.518267\pi\)
−0.0573552 + 0.998354i \(0.518267\pi\)
\(84\) −0.899030 + 2.79987i −0.0980922 + 0.305490i
\(85\) 11.5520 − 3.75072i 1.25299 − 0.406822i
\(86\) −5.87302 − 4.73907i −0.633304 − 0.511027i
\(87\) −1.87573 −0.201099
\(88\) −6.45430 − 3.24653i −0.688030 − 0.346081i
\(89\) 7.94252 + 4.58561i 0.841905 + 0.486074i 0.857911 − 0.513798i \(-0.171762\pi\)
−0.0160063 + 0.999872i \(0.505095\pi\)
\(90\) 3.18074 − 8.32381i 0.335280 − 0.877407i
\(91\) 10.6444 − 18.4367i 1.11584 − 1.93269i
\(92\) 1.47985 + 6.84557i 0.154285 + 0.713700i
\(93\) 1.24332 − 0.717830i 0.128926 − 0.0744355i
\(94\) −2.36214 + 15.0828i −0.243636 + 1.55568i
\(95\) 8.36548 + 5.00187i 0.858281 + 0.513181i
\(96\) 1.72186 + 1.69231i 0.175736 + 0.172721i
\(97\) −4.94059 − 8.55736i −0.501641 − 0.868868i −0.999998 − 0.00189613i \(-0.999396\pi\)
0.498357 − 0.866972i \(-0.333937\pi\)
\(98\) −6.42393 + 2.47868i −0.648915 + 0.250384i
\(99\) −6.23348 − 3.59890i −0.626488 − 0.361703i
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 380.2.s.a.259.8 yes 112
4.3 odd 2 inner 380.2.s.a.259.46 yes 112
5.4 even 2 inner 380.2.s.a.259.49 yes 112
19.8 odd 6 inner 380.2.s.a.179.11 yes 112
20.19 odd 2 inner 380.2.s.a.259.11 yes 112
76.27 even 6 inner 380.2.s.a.179.49 yes 112
95.84 odd 6 inner 380.2.s.a.179.46 yes 112
380.179 even 6 inner 380.2.s.a.179.8 ✓ 112
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.s.a.179.8 ✓ 112 380.179 even 6 inner
380.2.s.a.179.11 yes 112 19.8 odd 6 inner
380.2.s.a.179.46 yes 112 95.84 odd 6 inner
380.2.s.a.179.49 yes 112 76.27 even 6 inner
380.2.s.a.259.8 yes 112 1.1 even 1 trivial
380.2.s.a.259.11 yes 112 20.19 odd 2 inner
380.2.s.a.259.46 yes 112 4.3 odd 2 inner
380.2.s.a.259.49 yes 112 5.4 even 2 inner