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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 179.8
Character \(\chi\) \(=\) 380.179
Dual form 380.2.s.a.259.8

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

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

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 0.602888 − 0.797826i \(-0.294017\pi\)
−0.389494 + 0.921029i \(0.627350\pi\)
\(4\) 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 \(-0.874173\pi\)
0.922882 − 0.385084i \(-0.125827\pi\)
\(12\) 0.260958 + 0.812706i 0.0753320 + 0.234608i
\(13\) −3.08972 − 5.35155i −0.856934 − 1.48425i −0.874840 − 0.484413i \(-0.839033\pi\)
0.0179060 − 0.999840i \(-0.494300\pi\)
\(14\) 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 0.946648 − 0.322268i \(-0.104445\pi\)
0.194232 + 0.980956i \(0.437779\pi\)
\(18\) 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 0.623698 − 0.781665i \(-0.285629\pi\)
−0.988791 + 0.149306i \(0.952296\pi\)
\(24\) 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 −0.809871 − 0.586608i \(-0.800463\pi\)
0.103082 + 0.994673i \(0.467130\pi\)
\(30\) −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 0.0722146 − 0.997389i \(-0.476993\pi\)
−0.827657 + 0.561234i \(0.810327\pi\)
\(42\) 1.30579 + 1.61823i 0.201487 + 0.249698i
\(43\) 2.66813 − 4.62133i 0.406885 − 0.704746i −0.587653 − 0.809113i \(-0.699948\pi\)
0.994539 + 0.104366i \(0.0332815\pi\)
\(44\) 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 0.927603 + 0.373566i \(0.121865\pi\)
−0.140284 + 0.990111i \(0.544801\pi\)
\(48\) −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 0.705184 − 0.709024i \(-0.250864\pi\)
−0.966625 + 0.256195i \(0.917531\pi\)
\(54\) −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 0.130325 + 0.991471i \(0.458398\pi\)
−0.923802 + 0.382871i \(0.874935\pi\)
\(60\) 0.948955 + 1.65603i 0.122510 + 0.213793i
\(61\) −4.11002 − 7.11877i −0.526234 − 0.911465i −0.999533 − 0.0305625i \(-0.990270\pi\)
0.473299 − 0.880902i \(-0.343063\pi\)
\(62\) −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 −0.463664 − 0.886011i \(-0.653465\pi\)
0.535477 + 0.844550i \(0.320132\pi\)
\(68\) 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 −0.993675 − 0.112294i \(-0.964180\pi\)
0.594087 + 0.804401i \(0.297514\pi\)
\(72\) −4.37547 + 6.66165i −0.515654 + 0.785083i
\(73\) 3.37686 + 1.94963i 0.395232 + 0.228187i 0.684425 − 0.729084i \(-0.260053\pi\)
−0.289193 + 0.957271i \(0.593387\pi\)
\(74\) −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 −0.538780 − 0.842446i \(-0.681115\pi\)
0.998970 + 0.0453742i \(0.0144480\pi\)
\(80\) 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 −0.0160063 − 0.999872i \(-0.505095\pi\)
0.857911 + 0.513798i \(0.171762\pi\)
\(90\) 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 0.498357 + 0.866972i \(0.333937\pi\)
−0.999998 + 0.00189613i \(0.999396\pi\)
\(98\) −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.179.8 ✓ 112
4.3 odd 2 inner 380.2.s.a.179.46 yes 112
5.4 even 2 inner 380.2.s.a.179.49 yes 112
19.12 odd 6 inner 380.2.s.a.259.11 yes 112
20.19 odd 2 inner 380.2.s.a.179.11 yes 112
76.31 even 6 inner 380.2.s.a.259.49 yes 112
95.69 odd 6 inner 380.2.s.a.259.46 yes 112
380.259 even 6 inner 380.2.s.a.259.8 yes 112
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.s.a.179.8 ✓ 112 1.1 even 1 trivial
380.2.s.a.179.11 yes 112 20.19 odd 2 inner
380.2.s.a.179.46 yes 112 4.3 odd 2 inner
380.2.s.a.179.49 yes 112 5.4 even 2 inner
380.2.s.a.259.8 yes 112 380.259 even 6 inner
380.2.s.a.259.11 yes 112 19.12 odd 6 inner
380.2.s.a.259.46 yes 112 95.69 odd 6 inner
380.2.s.a.259.49 yes 112 76.31 even 6 inner