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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 7.6
Character \(\chi\) \(=\) 380.7
Dual form 380.2.v.c.163.6

$q$-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+(-1.37248 + 0.341027i) q^{2} +(-0.280019 + 1.04505i) q^{3} +(1.76740 - 0.936104i) q^{4} +(1.80654 + 1.31773i) q^{5} +(0.0279322 - 1.52980i) q^{6} +(3.16094 - 3.16094i) q^{7} +(-2.10649 + 1.88752i) q^{8} +(1.58437 + 0.914735i) q^{9} +(-2.92882 - 1.19249i) q^{10} -0.437437i q^{11} +(0.483365 + 2.10914i) q^{12} +(-1.27670 - 4.76472i) q^{13} +(-3.26036 + 5.41629i) q^{14} +(-1.88296 + 1.51892i) q^{15} +(2.24742 - 3.30894i) q^{16} +(1.14649 - 4.27877i) q^{17} +(-2.48646 - 0.715144i) q^{18} +(1.42657 - 4.11885i) q^{19} +(4.42641 + 0.637858i) q^{20} +(2.41820 + 4.18844i) q^{21} +(0.149178 + 0.600373i) q^{22} +(-5.89690 + 1.58007i) q^{23} +(-1.38268 - 2.72991i) q^{24} +(1.52715 + 4.76107i) q^{25} +(3.37714 + 6.10409i) q^{26} +(-3.69467 + 3.69467i) q^{27} +(2.62768 - 8.54561i) q^{28} +(7.30150 + 4.21552i) q^{29} +(2.06633 - 2.72683i) q^{30} +5.09810i q^{31} +(-1.95610 + 5.30789i) q^{32} +(0.457141 + 0.122491i) q^{33} +(-0.114364 + 6.26352i) q^{34} +(9.87562 - 1.54508i) q^{35} +(3.65650 + 0.133571i) q^{36} +(0.393671 + 0.393671i) q^{37} +(-0.553307 + 6.13953i) q^{38} +5.33684 q^{39} +(-6.29269 + 0.634078i) q^{40} +(-3.07132 - 5.31968i) q^{41} +(-4.74730 - 4.92388i) q^{42} +(-1.34149 + 5.00650i) q^{43} +(-0.409486 - 0.773126i) q^{44} +(1.65684 + 3.74028i) q^{45} +(7.55453 - 4.17962i) q^{46} +(-0.159082 - 0.593701i) q^{47} +(2.82868 + 3.27522i) q^{48} -12.9830i q^{49} +(-3.71964 - 6.01367i) q^{50} +(4.15047 + 2.39628i) q^{51} +(-6.71672 - 7.22604i) q^{52} +(0.558679 + 2.08502i) q^{53} +(3.81088 - 6.33084i) q^{54} +(0.576425 - 0.790246i) q^{55} +(-0.692153 + 12.6248i) q^{56} +(3.90491 + 2.64419i) q^{57} +(-11.4588 - 3.29571i) q^{58} +(4.33147 + 7.50233i) q^{59} +(-1.90607 + 4.44719i) q^{60} +(-7.13477 + 12.3578i) q^{61} +(-1.73859 - 6.99704i) q^{62} +(7.89951 - 2.11667i) q^{63} +(0.874570 - 7.95205i) q^{64} +(3.97222 - 10.2900i) q^{65} +(-0.669189 - 0.0122185i) q^{66} +(1.44106 + 5.37812i) q^{67} +(-1.97906 - 8.63555i) q^{68} -6.60498i q^{69} +(-13.0272 + 5.48844i) q^{70} +(-7.25049 + 4.18607i) q^{71} +(-5.06403 + 1.06364i) q^{72} +(-8.26542 - 2.21471i) q^{73} +(-0.674558 - 0.406053i) q^{74} +(-5.40317 + 0.262754i) q^{75} +(-1.33434 - 8.61508i) q^{76} +(-1.38271 - 1.38271i) q^{77} +(-7.32471 + 1.82001i) q^{78} +(-4.85533 - 8.40968i) q^{79} +(8.42035 - 3.01623i) q^{80} +(-0.0823140 - 0.142572i) q^{81} +(6.02947 + 6.25375i) q^{82} +(-0.249919 - 0.249919i) q^{83} +(8.19475 + 5.13897i) q^{84} +(7.70947 - 6.21899i) q^{85} +(0.133815 - 7.32880i) q^{86} +(-6.44997 + 6.44997i) q^{87} +(0.825668 + 0.921454i) q^{88} +(-9.89437 - 5.71252i) q^{89} +(-3.54952 - 4.56843i) q^{90} +(-19.0965 - 11.0254i) q^{91} +(-8.94308 + 8.31274i) q^{92} +(-5.32774 - 1.42756i) q^{93} +(0.420804 + 0.760591i) q^{94} +(8.00470 - 5.56100i) q^{95} +(-4.99924 - 3.53052i) q^{96} +(1.38295 - 5.16123i) q^{97} +(4.42756 + 17.8190i) q^{98} +(0.400139 - 0.693060i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.37248 + 0.341027i −0.970490 + 0.241142i
\(3\) −0.280019 + 1.04505i −0.161669 + 0.603357i 0.836773 + 0.547551i \(0.184440\pi\)
−0.998442 + 0.0558064i \(0.982227\pi\)
\(4\) 1.76740 − 0.936104i 0.883701 − 0.468052i
\(5\) 1.80654 + 1.31773i 0.807908 + 0.589309i
\(6\) 0.0279322 − 1.52980i 0.0114033 − 0.624537i
\(7\) 3.16094 − 3.16094i 1.19472 − 1.19472i 0.218996 − 0.975726i \(-0.429722\pi\)
0.975726 − 0.218996i \(-0.0702782\pi\)
\(8\) −2.10649 + 1.88752i −0.744755 + 0.667338i
\(9\) 1.58437 + 0.914735i 0.528123 + 0.304912i
\(10\) −2.92882 − 1.19249i −0.926174 − 0.377097i
\(11\) − 0.437437i − 0.131892i −0.997823 − 0.0659461i \(-0.978993\pi\)
0.997823 − 0.0659461i \(-0.0210065\pi\)
\(12\) 0.483365 + 2.10914i 0.139536 + 0.608857i
\(13\) −1.27670 − 4.76472i −0.354093 − 1.32149i −0.881622 − 0.471957i \(-0.843548\pi\)
0.527528 − 0.849538i \(-0.323119\pi\)
\(14\) −3.26036 + 5.41629i −0.871367 + 1.44756i
\(15\) −1.88296 + 1.51892i −0.486177 + 0.392184i
\(16\) 2.24742 − 3.30894i 0.561854 − 0.827236i
\(17\) 1.14649 − 4.27877i 0.278066 − 1.03776i −0.675693 − 0.737183i \(-0.736156\pi\)
0.953759 − 0.300572i \(-0.0971777\pi\)
\(18\) −2.48646 − 0.715144i −0.586065 − 0.168561i
\(19\) 1.42657 − 4.11885i 0.327278 − 0.944928i
\(20\) 4.42641 + 0.637858i 0.989776 + 0.142629i
\(21\) 2.41820 + 4.18844i 0.527694 + 0.913993i
\(22\) 0.149178 + 0.600373i 0.0318048 + 0.128000i
\(23\) −5.89690 + 1.58007i −1.22959 + 0.329467i −0.814419 − 0.580277i \(-0.802944\pi\)
−0.415170 + 0.909744i \(0.636278\pi\)
\(24\) −1.38268 − 2.72991i −0.282239 − 0.557241i
\(25\) 1.52715 + 4.76107i 0.305431 + 0.952214i
\(26\) 3.37714 + 6.10409i 0.662312 + 1.19711i
\(27\) −3.69467 + 3.69467i −0.711040 + 0.711040i
\(28\) 2.62768 − 8.54561i 0.496584 − 1.61497i
\(29\) 7.30150 + 4.21552i 1.35585 + 0.782803i 0.989062 − 0.147499i \(-0.0471225\pi\)
0.366793 + 0.930303i \(0.380456\pi\)
\(30\) 2.06633 − 2.72683i 0.377258 − 0.497848i
\(31\) 5.09810i 0.915646i 0.889043 + 0.457823i \(0.151371\pi\)
−0.889043 + 0.457823i \(0.848629\pi\)
\(32\) −1.95610 + 5.30789i −0.345792 + 0.938311i
\(33\) 0.457141 + 0.122491i 0.0795780 + 0.0213229i
\(34\) −0.114364 + 6.26352i −0.0196132 + 1.07418i
\(35\) 9.87562 − 1.54508i 1.66929 − 0.261165i
\(36\) 3.65650 + 0.133571i 0.609417 + 0.0222618i
\(37\) 0.393671 + 0.393671i 0.0647191 + 0.0647191i 0.738726 − 0.674006i \(-0.235428\pi\)
−0.674006 + 0.738726i \(0.735428\pi\)
\(38\) −0.553307 + 6.13953i −0.0897583 + 0.995964i
\(39\) 5.33684 0.854579
\(40\) −6.29269 + 0.634078i −0.994962 + 0.100257i
\(41\) −3.07132 − 5.31968i −0.479659 − 0.830794i 0.520069 − 0.854125i \(-0.325906\pi\)
−0.999728 + 0.0233303i \(0.992573\pi\)
\(42\) −4.74730 − 4.92388i −0.732524 − 0.759772i
\(43\) −1.34149 + 5.00650i −0.204575 + 0.763484i 0.785004 + 0.619491i \(0.212661\pi\)
−0.989579 + 0.143993i \(0.954006\pi\)
\(44\) −0.409486 − 0.773126i −0.0617324 − 0.116553i
\(45\) 1.65684 + 3.74028i 0.246987 + 0.557568i
\(46\) 7.55453 − 4.17962i 1.11386 − 0.616251i
\(47\) −0.159082 − 0.593701i −0.0232044 − 0.0866002i 0.953353 − 0.301859i \(-0.0976072\pi\)
−0.976557 + 0.215259i \(0.930941\pi\)
\(48\) 2.82868 + 3.27522i 0.408284 + 0.472737i
\(49\) − 12.9830i − 1.85472i
\(50\) −3.71964 − 6.01367i −0.526037 − 0.850462i
\(51\) 4.15047 + 2.39628i 0.581182 + 0.335546i
\(52\) −6.71672 − 7.22604i −0.931441 − 1.00207i
\(53\) 0.558679 + 2.08502i 0.0767404 + 0.286399i 0.993622 − 0.112759i \(-0.0359690\pi\)
−0.916882 + 0.399159i \(0.869302\pi\)
\(54\) 3.81088 − 6.33084i 0.518595 − 0.861518i
\(55\) 0.576425 − 0.790246i 0.0777252 − 0.106557i
\(56\) −0.692153 + 12.6248i −0.0924929 + 1.68706i
\(57\) 3.90491 + 2.64419i 0.517218 + 0.350231i
\(58\) −11.4588 − 3.29571i −1.50461 − 0.432749i
\(59\) 4.33147 + 7.50233i 0.563909 + 0.976720i 0.997150 + 0.0754416i \(0.0240367\pi\)
−0.433241 + 0.901278i \(0.642630\pi\)
\(60\) −1.90607 + 4.44719i −0.246073 + 0.574130i
\(61\) −7.13477 + 12.3578i −0.913514 + 1.58225i −0.104452 + 0.994530i \(0.533309\pi\)
−0.809062 + 0.587723i \(0.800024\pi\)
\(62\) −1.73859 − 6.99704i −0.220801 − 0.888625i
\(63\) 7.89951 − 2.11667i 0.995244 − 0.266675i
\(64\) 0.874570 − 7.95205i 0.109321 − 0.994006i
\(65\) 3.97222 − 10.2900i 0.492693 − 1.27632i
\(66\) −0.669189 − 0.0122185i −0.0823715 − 0.00150400i
\(67\) 1.44106 + 5.37812i 0.176054 + 0.657042i 0.996370 + 0.0851315i \(0.0271310\pi\)
−0.820316 + 0.571911i \(0.806202\pi\)
\(68\) −1.97906 − 8.63555i −0.239997 − 1.04721i
\(69\) − 6.60498i − 0.795146i
\(70\) −13.0272 + 5.48844i −1.55705 + 0.655994i
\(71\) −7.25049 + 4.18607i −0.860475 + 0.496795i −0.864171 − 0.503198i \(-0.832157\pi\)
0.00369633 + 0.999993i \(0.498823\pi\)
\(72\) −5.06403 + 1.06364i −0.596801 + 0.125351i
\(73\) −8.26542 − 2.21471i −0.967394 − 0.259212i −0.259667 − 0.965698i \(-0.583613\pi\)
−0.707727 + 0.706486i \(0.750279\pi\)
\(74\) −0.674558 − 0.406053i −0.0784158 − 0.0472027i
\(75\) −5.40317 + 0.262754i −0.623904 + 0.0303402i
\(76\) −1.33434 − 8.61508i −0.153059 − 0.988217i
\(77\) −1.38271 − 1.38271i −0.157574 − 0.157574i
\(78\) −7.32471 + 1.82001i −0.829360 + 0.206075i
\(79\) −4.85533 − 8.40968i −0.546268 − 0.946163i −0.998526 − 0.0542763i \(-0.982715\pi\)
0.452258 − 0.891887i \(-0.350618\pi\)
\(80\) 8.42035 − 3.01623i 0.941424 − 0.337225i
\(81\) −0.0823140 − 0.142572i −0.00914600 − 0.0158413i
\(82\) 6.02947 + 6.25375i 0.665844 + 0.690611i
\(83\) −0.249919 − 0.249919i −0.0274322 − 0.0274322i 0.693258 − 0.720690i \(-0.256175\pi\)
−0.720690 + 0.693258i \(0.756175\pi\)
\(84\) 8.19475 + 5.13897i 0.894120 + 0.560708i
\(85\) 7.70947 − 6.21899i 0.836209 − 0.674544i
\(86\) 0.133815 − 7.32880i 0.0144296 − 0.790285i
\(87\) −6.44997 + 6.44997i −0.691510 + 0.691510i
\(88\) 0.825668 + 0.921454i 0.0880165 + 0.0982274i
\(89\) −9.89437 − 5.71252i −1.04880 − 0.605525i −0.126487 − 0.991968i \(-0.540370\pi\)
−0.922313 + 0.386443i \(0.873704\pi\)
\(90\) −3.54952 − 4.56843i −0.374152 − 0.481555i
\(91\) −19.0965 − 11.0254i −2.00186 − 1.15578i
\(92\) −8.94308 + 8.31274i −0.932381 + 0.866663i
\(93\) −5.32774 − 1.42756i −0.552461 − 0.148032i
\(94\) 0.420804 + 0.760591i 0.0434026 + 0.0784490i
\(95\) 8.00470 − 5.56100i 0.821265 − 0.570547i
\(96\) −4.99924 − 3.53052i −0.510233 − 0.360332i
\(97\) 1.38295 − 5.16123i 0.140417 − 0.524043i −0.859500 − 0.511136i \(-0.829225\pi\)
0.999917 − 0.0129072i \(-0.00410860\pi\)
\(98\) 4.42756 + 17.8190i 0.447251 + 1.79999i
\(99\) 0.400139 − 0.693060i 0.0402154 − 0.0696552i
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.7.6 ✓ 216
4.3 odd 2 inner 380.2.v.c.7.41 yes 216
5.3 odd 4 inner 380.2.v.c.83.30 yes 216
19.11 even 3 inner 380.2.v.c.87.40 yes 216
20.3 even 4 inner 380.2.v.c.83.40 yes 216
76.11 odd 6 inner 380.2.v.c.87.30 yes 216
95.68 odd 12 inner 380.2.v.c.163.41 yes 216
380.163 even 12 inner 380.2.v.c.163.6 yes 216
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.v.c.7.6 ✓ 216 1.1 even 1 trivial
380.2.v.c.7.41 yes 216 4.3 odd 2 inner
380.2.v.c.83.30 yes 216 5.3 odd 4 inner
380.2.v.c.83.40 yes 216 20.3 even 4 inner
380.2.v.c.87.30 yes 216 76.11 odd 6 inner
380.2.v.c.87.40 yes 216 19.11 even 3 inner
380.2.v.c.163.6 yes 216 380.163 even 12 inner
380.2.v.c.163.41 yes 216 95.68 odd 12 inner