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Properties

Label 380.2.bh.a.13.3
Level $380$
Weight $2$
Character 380.13
Analytic conductor $3.034$
Analytic rank $0$
Dimension $120$
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] = [380,2,Mod(13,380)] 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("380.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([0, 27, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.bh (of order \(36\), degree \(12\), 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: \(120\)
Relative dimension: \(10\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 13.3
Character \(\chi\) \(=\) 380.13
Dual form 380.2.bh.a.117.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.41596 + 0.660272i) q^{3} +(-0.949889 + 2.02428i) q^{5} +(-1.12143 - 4.18524i) q^{7} +(-0.359387 + 0.428300i) q^{9} +(-0.627484 - 1.08683i) q^{11} +(1.92393 - 4.12588i) q^{13} +(0.00842766 - 3.49348i) q^{15} +(0.0727101 - 0.831080i) q^{17} +(3.83747 - 2.06733i) q^{19} +(4.35129 + 5.18567i) q^{21} +(0.587706 - 0.411516i) q^{23} +(-3.19542 - 3.84568i) q^{25} +(1.43917 - 5.37105i) q^{27} +(-3.22929 - 2.70969i) q^{29} +(-6.47928 - 3.74082i) q^{31} +(1.60610 + 1.12460i) q^{33} +(9.53733 + 1.70542i) q^{35} +(1.31792 + 1.31792i) q^{37} +7.11239i q^{39} +(-2.83535 + 7.79006i) q^{41} +(6.72347 - 9.60211i) q^{43} +(-0.525622 - 1.13434i) q^{45} +(-9.51062 + 0.832071i) q^{47} +(-10.1964 + 5.88691i) q^{49} +(0.445784 + 1.22478i) q^{51} +(-5.49098 - 7.84194i) q^{53} +(2.79610 - 0.237831i) q^{55} +(-4.06869 + 5.46102i) q^{57} +(-2.93676 + 2.46424i) q^{59} +(-1.31237 + 7.44285i) q^{61} +(2.19557 + 1.02381i) q^{63} +(6.52442 + 7.81370i) q^{65} +(0.616682 + 7.04871i) q^{67} +(-0.560454 + 0.970734i) q^{69} +(11.6384 - 2.05217i) q^{71} +(1.19747 + 2.56798i) q^{73} +(7.06378 + 3.33548i) q^{75} +(-3.84498 + 3.84498i) q^{77} +(-5.16648 - 1.88045i) q^{79} +(1.21729 + 6.90358i) q^{81} +(7.22139 - 1.93497i) q^{83} +(1.61327 + 0.936620i) q^{85} +(6.36167 + 1.70460i) q^{87} +(3.31376 - 1.20611i) q^{89} +(-19.4253 - 3.42521i) q^{91} +(11.6443 + 1.01875i) q^{93} +(0.539677 + 9.73184i) q^{95} +(-5.44825 - 0.476660i) q^{97} +(0.691001 + 0.121842i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 6 q^{7} + 18 q^{15} - 18 q^{17} + 48 q^{21} - 36 q^{23} - 24 q^{25} - 60 q^{33} - 18 q^{35} - 12 q^{41} - 36 q^{43} + 18 q^{45} - 24 q^{47} + 96 q^{51} - 18 q^{53} + 72 q^{55} - 6 q^{57} - 24 q^{61}+ \cdots - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.41596 + 0.660272i −0.817503 + 0.381208i −0.785947 − 0.618294i \(-0.787824\pi\)
−0.0315563 + 0.999502i \(0.510046\pi\)
\(4\) 0 0
\(5\) −0.949889 + 2.02428i −0.424803 + 0.905286i
\(6\) 0 0
\(7\) −1.12143 − 4.18524i −0.423861 − 1.58187i −0.766397 − 0.642367i \(-0.777953\pi\)
0.342536 − 0.939505i \(-0.388714\pi\)
\(8\) 0 0
\(9\) −0.359387 + 0.428300i −0.119796 + 0.142767i
\(10\) 0 0
\(11\) −0.627484 − 1.08683i −0.189194 − 0.327693i 0.755788 − 0.654816i \(-0.227254\pi\)
−0.944982 + 0.327123i \(0.893921\pi\)
\(12\) 0 0
\(13\) 1.92393 − 4.12588i 0.533602 − 1.14431i −0.436072 − 0.899912i \(-0.643631\pi\)
0.969674 − 0.244402i \(-0.0785915\pi\)
\(14\) 0 0
\(15\) 0.00842766 − 3.49348i 0.00217601 − 0.902012i
\(16\) 0 0
\(17\) 0.0727101 − 0.831080i 0.0176348 − 0.201567i −0.982266 − 0.187492i \(-0.939964\pi\)
0.999901 − 0.0140747i \(-0.00448027\pi\)
\(18\) 0 0
\(19\) 3.83747 − 2.06733i 0.880376 − 0.474277i
\(20\) 0 0
\(21\) 4.35129 + 5.18567i 0.949530 + 1.13161i
\(22\) 0 0
\(23\) 0.587706 − 0.411516i 0.122545 − 0.0858070i −0.510697 − 0.859761i \(-0.670613\pi\)
0.633242 + 0.773954i \(0.281724\pi\)
\(24\) 0 0
\(25\) −3.19542 − 3.84568i −0.639084 − 0.769137i
\(26\) 0 0
\(27\) 1.43917 − 5.37105i 0.276968 − 1.03366i
\(28\) 0 0
\(29\) −3.22929 − 2.70969i −0.599664 − 0.503177i 0.291674 − 0.956518i \(-0.405788\pi\)
−0.891337 + 0.453340i \(0.850232\pi\)
\(30\) 0 0
\(31\) −6.47928 − 3.74082i −1.16371 − 0.671870i −0.211522 − 0.977373i \(-0.567842\pi\)
−0.952191 + 0.305503i \(0.901175\pi\)
\(32\) 0 0
\(33\) 1.60610 + 1.12460i 0.279586 + 0.195768i
\(34\) 0 0
\(35\) 9.53733 + 1.70542i 1.61210 + 0.288269i
\(36\) 0 0
\(37\) 1.31792 + 1.31792i 0.216665 + 0.216665i 0.807091 − 0.590427i \(-0.201040\pi\)
−0.590427 + 0.807091i \(0.701040\pi\)
\(38\) 0 0
\(39\) 7.11239i 1.13889i
\(40\) 0 0
\(41\) −2.83535 + 7.79006i −0.442807 + 1.21660i 0.494831 + 0.868989i \(0.335230\pi\)
−0.937638 + 0.347613i \(0.886992\pi\)
\(42\) 0 0
\(43\) 6.72347 − 9.60211i 1.02532 − 1.46431i 0.145290 − 0.989389i \(-0.453589\pi\)
0.880030 − 0.474919i \(-0.157523\pi\)
\(44\) 0 0
\(45\) −0.525622 − 1.13434i −0.0783552 − 0.169097i
\(46\) 0 0
\(47\) −9.51062 + 0.832071i −1.38727 + 0.121370i −0.756191 − 0.654351i \(-0.772942\pi\)
−0.631076 + 0.775721i \(0.717386\pi\)
\(48\) 0 0
\(49\) −10.1964 + 5.88691i −1.45663 + 0.840988i
\(50\) 0 0
\(51\) 0.445784 + 1.22478i 0.0624223 + 0.171504i
\(52\) 0 0
\(53\) −5.49098 − 7.84194i −0.754245 − 1.07717i −0.994321 − 0.106419i \(-0.966062\pi\)
0.240077 − 0.970754i \(-0.422827\pi\)
\(54\) 0 0
\(55\) 2.79610 − 0.237831i 0.377026 − 0.0320692i
\(56\) 0 0
\(57\) −4.06869 + 5.46102i −0.538912 + 0.723329i
\(58\) 0 0
\(59\) −2.93676 + 2.46424i −0.382334 + 0.320816i −0.813618 − 0.581400i \(-0.802505\pi\)
0.431284 + 0.902216i \(0.358061\pi\)
\(60\) 0 0
\(61\) −1.31237 + 7.44285i −0.168032 + 0.952959i 0.777850 + 0.628450i \(0.216310\pi\)
−0.945883 + 0.324509i \(0.894801\pi\)
\(62\) 0 0
\(63\) 2.19557 + 1.02381i 0.276615 + 0.128988i
\(64\) 0 0
\(65\) 6.52442 + 7.81370i 0.809254 + 0.969170i
\(66\) 0 0
\(67\) 0.616682 + 7.04871i 0.0753398 + 0.861137i 0.936010 + 0.351974i \(0.114489\pi\)
−0.860670 + 0.509163i \(0.829955\pi\)
\(68\) 0 0
\(69\) −0.560454 + 0.970734i −0.0674707 + 0.116863i
\(70\) 0 0
\(71\) 11.6384 − 2.05217i 1.38123 − 0.243548i 0.566821 − 0.823841i \(-0.308173\pi\)
0.814408 + 0.580293i \(0.197062\pi\)
\(72\) 0 0
\(73\) 1.19747 + 2.56798i 0.140153 + 0.300560i 0.963838 − 0.266488i \(-0.0858632\pi\)
−0.823685 + 0.567048i \(0.808085\pi\)
\(74\) 0 0
\(75\) 7.06378 + 3.33548i 0.815655 + 0.385148i
\(76\) 0 0
\(77\) −3.84498 + 3.84498i −0.438176 + 0.438176i
\(78\) 0 0
\(79\) −5.16648 − 1.88045i −0.581275 − 0.211567i 0.0346128 − 0.999401i \(-0.488980\pi\)
−0.615888 + 0.787834i \(0.711202\pi\)
\(80\) 0 0
\(81\) 1.21729 + 6.90358i 0.135254 + 0.767065i
\(82\) 0 0
\(83\) 7.22139 − 1.93497i 0.792651 − 0.212390i 0.160296 − 0.987069i \(-0.448755\pi\)
0.632355 + 0.774679i \(0.282088\pi\)
\(84\) 0 0
\(85\) 1.61327 + 0.936620i 0.174984 + 0.101591i
\(86\) 0 0
\(87\) 6.36167 + 1.70460i 0.682042 + 0.182753i
\(88\) 0 0
\(89\) 3.31376 − 1.20611i 0.351258 − 0.127847i −0.160364 − 0.987058i \(-0.551267\pi\)
0.511622 + 0.859210i \(0.329045\pi\)
\(90\) 0 0
\(91\) −19.4253 − 3.42521i −2.03633 − 0.359060i
\(92\) 0 0
\(93\) 11.6443 + 1.01875i 1.20746 + 0.105639i
\(94\) 0 0
\(95\) 0.539677 + 9.73184i 0.0553697 + 0.998466i
\(96\) 0 0
\(97\) −5.44825 − 0.476660i −0.553186 − 0.0483975i −0.192864 − 0.981225i \(-0.561778\pi\)
−0.360322 + 0.932828i \(0.617333\pi\)
\(98\) 0 0
\(99\) 0.691001 + 0.121842i 0.0694482 + 0.0122456i
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.bh.a.13.3 ✓ 120
5.2 odd 4 inner 380.2.bh.a.317.8 yes 120
19.3 odd 18 inner 380.2.bh.a.193.8 yes 120
95.22 even 36 inner 380.2.bh.a.117.3 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.13.3 ✓ 120 1.1 even 1 trivial
380.2.bh.a.117.3 yes 120 95.22 even 36 inner
380.2.bh.a.193.8 yes 120 19.3 odd 18 inner
380.2.bh.a.317.8 yes 120 5.2 odd 4 inner