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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,2,Mod(7,380)] 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("380.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.v (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [216,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(54\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 7.13
Character \(\chi\) \(=\) 380.7
Dual form 380.2.v.c.163.13

$q$-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+(-1.04438 - 0.953560i) q^{2} +(-0.712111 + 2.65764i) q^{3} +(0.181448 + 1.99175i) q^{4} +(1.75700 - 1.38309i) q^{5} +(3.27793 - 2.09653i) q^{6} +(2.28576 - 2.28576i) q^{7} +(1.70975 - 2.25316i) q^{8} +(-3.95785 - 2.28507i) q^{9} +(-3.15383 - 0.230937i) q^{10} -5.10396i q^{11} +(-5.42256 - 0.936127i) q^{12} +(0.779486 + 2.90908i) q^{13} +(-4.56680 + 0.207587i) q^{14} +(2.42457 + 5.65439i) q^{15} +(-3.93415 + 0.722798i) q^{16} +(-0.0673490 + 0.251350i) q^{17} +(1.95454 + 6.16052i) q^{18} +(1.58489 - 4.06056i) q^{19} +(3.07358 + 3.24855i) q^{20} +(4.44699 + 7.70242i) q^{21} +(-4.86693 + 5.33046i) q^{22} +(7.18875 - 1.92622i) q^{23} +(4.77055 + 6.14841i) q^{24} +(1.17411 - 4.86019i) q^{25} +(1.95991 - 3.78147i) q^{26} +(3.05473 - 3.05473i) q^{27} +(4.96740 + 4.13791i) q^{28} +(-3.30285 - 1.90690i) q^{29} +(2.85963 - 8.21729i) q^{30} +3.55626i q^{31} +(4.79797 + 2.99658i) q^{32} +(13.5645 + 3.63458i) q^{33} +(0.310015 - 0.198283i) q^{34} +(0.854668 - 7.17749i) q^{35} +(3.83314 - 8.29767i) q^{36} +(1.75278 + 1.75278i) q^{37} +(-5.52720 + 2.72947i) q^{38} -8.28636 q^{39} +(-0.112287 - 6.32356i) q^{40} +(0.664135 + 1.15032i) q^{41} +(2.70038 - 12.2847i) q^{42} +(-3.36983 + 12.5764i) q^{43} +(10.1658 - 0.926101i) q^{44} +(-10.1144 + 1.45920i) q^{45} +(-9.34453 - 4.84320i) q^{46} +(2.48266 + 9.26542i) q^{47} +(0.880622 - 10.9703i) q^{48} -3.44936i q^{49} +(-5.86070 + 3.95628i) q^{50} +(-0.620036 - 0.357978i) q^{51} +(-5.65273 + 2.08039i) q^{52} +(-1.99614 - 7.44969i) q^{53} +(-6.10315 + 0.277423i) q^{54} +(-7.05924 - 8.96766i) q^{55} +(-1.24210 - 9.05826i) q^{56} +(9.66287 + 7.10362i) q^{57} +(1.63108 + 5.14099i) q^{58} +(-0.751075 - 1.30090i) q^{59} +(-10.8222 + 5.85512i) q^{60} +(-0.805951 + 1.39595i) q^{61} +(3.39110 - 3.71407i) q^{62} +(-14.2698 + 3.82357i) q^{63} +(-2.15348 - 7.70471i) q^{64} +(5.39309 + 4.03316i) q^{65} +(-10.7006 - 16.7304i) q^{66} +(-0.934759 - 3.48857i) q^{67} +(-0.512847 - 0.0885356i) q^{68} +20.4768i q^{69} +(-7.73676 + 6.68103i) q^{70} +(5.93706 - 3.42776i) q^{71} +(-11.9156 + 5.01077i) q^{72} +(-0.106260 - 0.0284723i) q^{73} +(-0.159183 - 3.50195i) q^{74} +(12.0805 + 6.58137i) q^{75} +(8.37520 + 2.41992i) q^{76} +(-11.6664 - 11.6664i) q^{77} +(8.65409 + 7.90154i) q^{78} +(-5.86204 - 10.1534i) q^{79} +(-5.91262 + 6.71125i) q^{80} +(-0.912151 - 1.57989i) q^{81} +(0.403287 - 1.83466i) q^{82} +(-0.134588 - 0.134588i) q^{83} +(-14.5344 + 10.2549i) q^{84} +(0.229307 + 0.534772i) q^{85} +(15.5117 - 9.92115i) q^{86} +(7.41985 - 7.41985i) q^{87} +(-11.5000 - 8.72651i) q^{88} +(8.24527 + 4.76041i) q^{89} +(11.9547 + 8.12073i) q^{90} +(8.43116 + 4.86773i) q^{91} +(5.14094 + 13.9687i) q^{92} +(-9.45124 - 2.53245i) q^{93} +(6.24230 - 12.0440i) q^{94} +(-2.83148 - 9.32645i) q^{95} +(-11.3805 + 10.6174i) q^{96} +(-1.29206 + 4.82202i) q^{97} +(-3.28917 + 3.60243i) q^{98} +(-11.6629 + 20.2007i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 216 q + 12 q^{6} - 36 q^{8} + 4 q^{10} - 20 q^{12} - 8 q^{13} + 8 q^{16} - 16 q^{17} + 12 q^{18} - 48 q^{20} + 40 q^{21} + 16 q^{22} - 16 q^{25} + 24 q^{26} + 8 q^{28} - 12 q^{30} - 20 q^{32} + 20 q^{33}+ \cdots + 34 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.04438 − 0.953560i −0.738486 − 0.674269i
\(3\) −0.712111 + 2.65764i −0.411138 + 1.53439i 0.381310 + 0.924447i \(0.375473\pi\)
−0.792448 + 0.609940i \(0.791194\pi\)
\(4\) 0.181448 + 1.99175i 0.0907239 + 0.995876i
\(5\) 1.75700 − 1.38309i 0.785755 − 0.618537i
\(6\) 3.27793 − 2.09653i 1.33821 − 0.855906i
\(7\) 2.28576 − 2.28576i 0.863934 − 0.863934i −0.127858 − 0.991792i \(-0.540810\pi\)
0.991792 + 0.127858i \(0.0408102\pi\)
\(8\) 1.70975 − 2.25316i 0.604490 − 0.796613i
\(9\) −3.95785 − 2.28507i −1.31928 − 0.761688i
\(10\) −3.15383 − 0.230937i −0.997330 − 0.0730287i
\(11\) − 5.10396i − 1.53890i −0.638707 − 0.769450i \(-0.720530\pi\)
0.638707 − 0.769450i \(-0.279470\pi\)
\(12\) −5.42256 − 0.936127i −1.56536 − 0.270237i
\(13\) 0.779486 + 2.90908i 0.216191 + 0.806834i 0.985744 + 0.168252i \(0.0538122\pi\)
−0.769553 + 0.638582i \(0.779521\pi\)
\(14\) −4.56680 + 0.207587i −1.22053 + 0.0554799i
\(15\) 2.42457 + 5.65439i 0.626022 + 1.45996i
\(16\) −3.93415 + 0.722798i −0.983538 + 0.180699i
\(17\) −0.0673490 + 0.251350i −0.0163345 + 0.0609613i −0.973612 − 0.228210i \(-0.926713\pi\)
0.957278 + 0.289171i \(0.0933795\pi\)
\(18\) 1.95454 + 6.16052i 0.460690 + 1.45205i
\(19\) 1.58489 − 4.06056i 0.363598 − 0.931556i
\(20\) 3.07358 + 3.24855i 0.687273 + 0.726399i
\(21\) 4.44699 + 7.70242i 0.970413 + 1.68081i
\(22\) −4.86693 + 5.33046i −1.03763 + 1.13646i
\(23\) 7.18875 − 1.92622i 1.49896 − 0.401645i 0.586207 − 0.810161i \(-0.300620\pi\)
0.912751 + 0.408516i \(0.133954\pi\)
\(24\) 4.77055 + 6.14841i 0.973784 + 1.25504i
\(25\) 1.17411 − 4.86019i 0.234823 − 0.972038i
\(26\) 1.95991 − 3.78147i 0.384369 − 0.741606i
\(27\) 3.05473 − 3.05473i 0.587882 − 0.587882i
\(28\) 4.96740 + 4.13791i 0.938751 + 0.781992i
\(29\) −3.30285 − 1.90690i −0.613324 − 0.354103i 0.160941 − 0.986964i \(-0.448547\pi\)
−0.774265 + 0.632861i \(0.781880\pi\)
\(30\) 2.85963 − 8.21729i 0.522094 − 1.50026i
\(31\) 3.55626i 0.638723i 0.947633 + 0.319361i \(0.103468\pi\)
−0.947633 + 0.319361i \(0.896532\pi\)
\(32\) 4.79797 + 2.99658i 0.848170 + 0.529725i
\(33\) 13.5645 + 3.63458i 2.36127 + 0.632700i
\(34\) 0.310015 − 0.198283i 0.0531671 − 0.0340052i
\(35\) 0.854668 − 7.17749i 0.144465 − 1.21322i
\(36\) 3.83314 − 8.29767i 0.638857 − 1.38295i
\(37\) 1.75278 + 1.75278i 0.288156 + 0.288156i 0.836351 − 0.548195i \(-0.184685\pi\)
−0.548195 + 0.836351i \(0.684685\pi\)
\(38\) −5.52720 + 2.72947i −0.896631 + 0.442779i
\(39\) −8.28636 −1.32688
\(40\) −0.112287 − 6.32356i −0.0177541 − 0.999842i
\(41\) 0.664135 + 1.15032i 0.103720 + 0.179649i 0.913215 − 0.407479i \(-0.133592\pi\)
−0.809494 + 0.587128i \(0.800259\pi\)
\(42\) 2.70038 − 12.2847i 0.416677 − 1.89557i
\(43\) −3.36983 + 12.5764i −0.513895 + 1.91788i −0.140989 + 0.990011i \(0.545028\pi\)
−0.372905 + 0.927869i \(0.621638\pi\)
\(44\) 10.1658 − 0.926101i 1.53255 − 0.139615i
\(45\) −10.1144 + 1.45920i −1.50777 + 0.217525i
\(46\) −9.34453 − 4.84320i −1.37778 − 0.714091i
\(47\) 2.48266 + 9.26542i 0.362133 + 1.35150i 0.871265 + 0.490812i \(0.163300\pi\)
−0.509132 + 0.860688i \(0.670033\pi\)
\(48\) 0.880622 − 10.9703i 0.127107 − 1.58342i
\(49\) − 3.44936i − 0.492765i
\(50\) −5.86070 + 3.95628i −0.828828 + 0.559503i
\(51\) −0.620036 − 0.357978i −0.0868224 − 0.0501269i
\(52\) −5.65273 + 2.08039i −0.783893 + 0.288498i
\(53\) −1.99614 − 7.44969i −0.274191 − 1.02329i −0.956382 − 0.292120i \(-0.905639\pi\)
0.682191 − 0.731174i \(-0.261027\pi\)
\(54\) −6.10315 + 0.277423i −0.830534 + 0.0377525i
\(55\) −7.05924 − 8.96766i −0.951868 − 1.20920i
\(56\) −1.24210 − 9.05826i −0.165982 − 1.21046i
\(57\) 9.66287 + 7.10362i 1.27988 + 0.940897i
\(58\) 1.63108 + 5.14099i 0.214171 + 0.675046i
\(59\) −0.751075 − 1.30090i −0.0977816 − 0.169363i 0.812984 − 0.582285i \(-0.197841\pi\)
−0.910766 + 0.412923i \(0.864508\pi\)
\(60\) −10.8222 + 5.85512i −1.39714 + 0.755893i
\(61\) −0.805951 + 1.39595i −0.103191 + 0.178733i −0.912998 − 0.407964i \(-0.866239\pi\)
0.809806 + 0.586697i \(0.199572\pi\)
\(62\) 3.39110 − 3.71407i 0.430671 − 0.471688i
\(63\) −14.2698 + 3.82357i −1.79782 + 0.481725i
\(64\) −2.15348 − 7.70471i −0.269185 − 0.963089i
\(65\) 5.39309 + 4.03316i 0.668930 + 0.500252i
\(66\) −10.7006 − 16.7304i −1.31715 − 2.05937i
\(67\) −0.934759 − 3.48857i −0.114199 − 0.426197i 0.885027 − 0.465540i \(-0.154140\pi\)
−0.999226 + 0.0393437i \(0.987473\pi\)
\(68\) −0.512847 − 0.0885356i −0.0621918 − 0.0107365i
\(69\) 20.4768i 2.46511i
\(70\) −7.73676 + 6.68103i −0.924720 + 0.798535i
\(71\) 5.93706 − 3.42776i 0.704600 − 0.406801i −0.104459 − 0.994529i \(-0.533311\pi\)
0.809058 + 0.587728i \(0.199978\pi\)
\(72\) −11.9156 + 5.01077i −1.40426 + 0.590525i
\(73\) −0.106260 − 0.0284723i −0.0124368 − 0.00333243i 0.252595 − 0.967572i \(-0.418716\pi\)
−0.265032 + 0.964240i \(0.585383\pi\)
\(74\) −0.159183 − 3.50195i −0.0185047 − 0.407093i
\(75\) 12.0805 + 6.58137i 1.39494 + 0.759951i
\(76\) 8.37520 + 2.41992i 0.960701 + 0.277584i
\(77\) −11.6664 − 11.6664i −1.32951 − 1.32951i
\(78\) 8.65409 + 7.90154i 0.979882 + 0.894673i
\(79\) −5.86204 − 10.1534i −0.659531 − 1.14234i −0.980737 − 0.195332i \(-0.937422\pi\)
0.321206 − 0.947009i \(-0.395912\pi\)
\(80\) −5.91262 + 6.71125i −0.661051 + 0.750341i
\(81\) −0.912151 − 1.57989i −0.101350 − 0.175543i
\(82\) 0.403287 − 1.83466i 0.0445356 − 0.202604i
\(83\) −0.134588 − 0.134588i −0.0147730 − 0.0147730i 0.699682 − 0.714455i \(-0.253325\pi\)
−0.714455 + 0.699682i \(0.753325\pi\)
\(84\) −14.5344 + 10.2549i −1.58583 + 1.11890i
\(85\) 0.229307 + 0.534772i 0.0248719 + 0.0580041i
\(86\) 15.5117 − 9.92115i 1.67267 − 1.06983i
\(87\) 7.41985 − 7.41985i 0.795492 − 0.795492i
\(88\) −11.5000 − 8.72651i −1.22591 − 0.930249i
\(89\) 8.24527 + 4.76041i 0.873997 + 0.504602i 0.868674 − 0.495383i \(-0.164972\pi\)
0.00532255 + 0.999986i \(0.498306\pi\)
\(90\) 11.9547 + 8.12073i 1.26014 + 0.856000i
\(91\) 8.43116 + 4.86773i 0.883826 + 0.510277i
\(92\) 5.14094 + 13.9687i 0.535980 + 1.45634i
\(93\) −9.45124 − 2.53245i −0.980047 − 0.262603i
\(94\) 6.24230 − 12.0440i 0.643844 − 1.24224i
\(95\) −2.83148 − 9.32645i −0.290503 − 0.956874i
\(96\) −11.3805 + 10.6174i −1.16152 + 1.08363i
\(97\) −1.29206 + 4.82202i −0.131188 + 0.489602i −0.999984 − 0.00556811i \(-0.998228\pi\)
0.868796 + 0.495170i \(0.164894\pi\)
\(98\) −3.28917 + 3.60243i −0.332256 + 0.363900i
\(99\) −11.6629 + 20.2007i −1.17216 + 2.03025i
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.v.c.7.13 ✓ 216
4.3 odd 2 inner 380.2.v.c.7.52 yes 216
5.3 odd 4 inner 380.2.v.c.83.15 yes 216
19.11 even 3 inner 380.2.v.c.87.22 yes 216
20.3 even 4 inner 380.2.v.c.83.22 yes 216
76.11 odd 6 inner 380.2.v.c.87.15 yes 216
95.68 odd 12 inner 380.2.v.c.163.52 yes 216
380.163 even 12 inner 380.2.v.c.163.13 yes 216
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.v.c.7.13 ✓ 216 1.1 even 1 trivial
380.2.v.c.7.52 yes 216 4.3 odd 2 inner
380.2.v.c.83.15 yes 216 5.3 odd 4 inner
380.2.v.c.83.22 yes 216 20.3 even 4 inner
380.2.v.c.87.15 yes 216 76.11 odd 6 inner
380.2.v.c.87.22 yes 216 19.11 even 3 inner
380.2.v.c.163.13 yes 216 380.163 even 12 inner
380.2.v.c.163.52 yes 216 95.68 odd 12 inner