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Properties

Label 380.2.bh.a.13.5
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 13.5
Character \(\chi\) \(=\) 380.13
Dual form 380.2.bh.a.117.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.362674 + 0.169118i) q^{3} +(1.57660 - 1.58566i) q^{5} +(1.00773 + 3.76091i) q^{7} +(-1.82543 + 2.17546i) q^{9} +(0.973938 + 1.68691i) q^{11} +(2.46032 - 5.27618i) q^{13} +(-0.303629 + 0.841710i) q^{15} +(-0.350620 + 4.00760i) q^{17} +(4.22053 + 1.08956i) q^{19} +(-1.00151 - 1.19356i) q^{21} +(1.79737 - 1.25853i) q^{23} +(-0.0286516 - 4.99992i) q^{25} +(0.604839 - 2.25729i) q^{27} +(3.44950 + 2.89447i) q^{29} +(2.99397 + 1.72857i) q^{31} +(-0.638508 - 0.447088i) q^{33} +(7.55232 + 4.33153i) q^{35} +(-5.01505 - 5.01505i) q^{37} +2.32961i q^{39} +(-3.17796 + 8.73136i) q^{41} +(-1.28086 + 1.82925i) q^{43} +(0.571574 + 6.32436i) q^{45} +(-4.06756 + 0.355865i) q^{47} +(-7.06672 + 4.07998i) q^{49} +(-0.550596 - 1.51275i) q^{51} +(-4.39667 - 6.27909i) q^{53} +(4.21038 + 1.11525i) q^{55} +(-1.71494 + 0.318611i) q^{57} +(4.29987 - 3.60802i) q^{59} +(1.94572 - 11.0347i) q^{61} +(-10.0213 - 4.67299i) q^{63} +(-4.48729 - 12.2197i) q^{65} +(-0.637363 - 7.28509i) q^{67} +(-0.439018 + 0.760402i) q^{69} +(2.69808 - 0.475744i) q^{71} +(-1.74492 - 3.74199i) q^{73} +(0.855965 + 1.80849i) q^{75} +(-5.36285 + 5.36285i) q^{77} +(-8.14836 - 2.96576i) q^{79} +(-1.31702 - 7.46922i) q^{81} +(-14.3797 + 3.85303i) q^{83} +(5.80192 + 6.87436i) q^{85} +(-1.74055 - 0.466379i) q^{87} +(-1.72328 + 0.627223i) q^{89} +(22.3226 + 3.93607i) q^{91} +(-1.37817 - 0.120574i) q^{93} +(8.38177 - 4.97453i) q^{95} +(-0.221785 - 0.0194037i) q^{97} +(-5.44767 - 0.960571i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 6 q^{7} + 18 q^{15} - 18 q^{17} + 48 q^{21} - 36 q^{23} - 24 q^{25} - 60 q^{33} - 18 q^{35} - 12 q^{41} - 36 q^{43} + 18 q^{45} - 24 q^{47} + 96 q^{51} - 18 q^{53} + 72 q^{55} - 6 q^{57} - 24 q^{61}+ \cdots - 60 q^{97}+O(q^{100}) \) 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
\(3\) −0.362674 + 0.169118i −0.209390 + 0.0976401i −0.524484 − 0.851420i \(-0.675742\pi\)
0.315094 + 0.949060i \(0.397964\pi\)
\(4\) 0 0
\(5\) 1.57660 − 1.58566i 0.705078 − 0.709130i
\(6\) 0 0
\(7\) 1.00773 + 3.76091i 0.380887 + 1.42149i 0.844549 + 0.535478i \(0.179868\pi\)
−0.463662 + 0.886012i \(0.653465\pi\)
\(8\) 0 0
\(9\) −1.82543 + 2.17546i −0.608477 + 0.725155i
\(10\) 0 0
\(11\) 0.973938 + 1.68691i 0.293653 + 0.508623i 0.974671 − 0.223645i \(-0.0717955\pi\)
−0.681017 + 0.732267i \(0.738462\pi\)
\(12\) 0 0
\(13\) 2.46032 − 5.27618i 0.682370 − 1.46335i −0.193649 − 0.981071i \(-0.562032\pi\)
0.876019 − 0.482277i \(-0.160190\pi\)
\(14\) 0 0
\(15\) −0.303629 + 0.841710i −0.0783967 + 0.217328i
\(16\) 0 0
\(17\) −0.350620 + 4.00760i −0.0850378 + 0.971986i 0.827087 + 0.562074i \(0.189996\pi\)
−0.912125 + 0.409913i \(0.865559\pi\)
\(18\) 0 0
\(19\) 4.22053 + 1.08956i 0.968256 + 0.249962i
\(20\) 0 0
\(21\) −1.00151 − 1.19356i −0.218548 − 0.260456i
\(22\) 0 0
\(23\) 1.79737 − 1.25853i 0.374777 − 0.262421i −0.370979 − 0.928641i \(-0.620978\pi\)
0.745755 + 0.666220i \(0.232089\pi\)
\(24\) 0 0
\(25\) −0.0286516 − 4.99992i −0.00573033 − 0.999984i
\(26\) 0 0
\(27\) 0.604839 − 2.25729i 0.116401 − 0.434416i
\(28\) 0 0
\(29\) 3.44950 + 2.89447i 0.640556 + 0.537490i 0.904189 − 0.427133i \(-0.140476\pi\)
−0.263633 + 0.964623i \(0.584921\pi\)
\(30\) 0 0
\(31\) 2.99397 + 1.72857i 0.537733 + 0.310460i 0.744160 − 0.668002i \(-0.232850\pi\)
−0.206427 + 0.978462i \(0.566184\pi\)
\(32\) 0 0
\(33\) −0.638508 − 0.447088i −0.111150 − 0.0778281i
\(34\) 0 0
\(35\) 7.55232 + 4.33153i 1.27658 + 0.732162i
\(36\) 0 0
\(37\) −5.01505 − 5.01505i −0.824469 − 0.824469i 0.162276 − 0.986745i \(-0.448116\pi\)
−0.986745 + 0.162276i \(0.948116\pi\)
\(38\) 0 0
\(39\) 2.32961i 0.373037i
\(40\) 0 0
\(41\) −3.17796 + 8.73136i −0.496313 + 1.36361i 0.398500 + 0.917168i \(0.369531\pi\)
−0.894813 + 0.446441i \(0.852691\pi\)
\(42\) 0 0
\(43\) −1.28086 + 1.82925i −0.195329 + 0.278959i −0.904854 − 0.425721i \(-0.860020\pi\)
0.709525 + 0.704680i \(0.248909\pi\)
\(44\) 0 0
\(45\) 0.571574 + 6.32436i 0.0852051 + 0.942780i
\(46\) 0 0
\(47\) −4.06756 + 0.355865i −0.593314 + 0.0519083i −0.379859 − 0.925045i \(-0.624027\pi\)
−0.213455 + 0.976953i \(0.568472\pi\)
\(48\) 0 0
\(49\) −7.06672 + 4.07998i −1.00953 + 0.582854i
\(50\) 0 0
\(51\) −0.550596 − 1.51275i −0.0770988 − 0.211827i
\(52\) 0 0
\(53\) −4.39667 − 6.27909i −0.603929 − 0.862500i 0.394479 − 0.918905i \(-0.370925\pi\)
−0.998408 + 0.0564053i \(0.982036\pi\)
\(54\) 0 0
\(55\) 4.21038 + 1.11525i 0.567728 + 0.150380i
\(56\) 0 0
\(57\) −1.71494 + 0.318611i −0.227149 + 0.0422010i
\(58\) 0 0
\(59\) 4.29987 − 3.60802i 0.559795 − 0.469724i −0.318447 − 0.947941i \(-0.603161\pi\)
0.878242 + 0.478217i \(0.158717\pi\)
\(60\) 0 0
\(61\) 1.94572 − 11.0347i 0.249124 − 1.41285i −0.561592 − 0.827414i \(-0.689811\pi\)
0.810716 − 0.585440i \(-0.199078\pi\)
\(62\) 0 0
\(63\) −10.0213 − 4.67299i −1.26256 − 0.588742i
\(64\) 0 0
\(65\) −4.48729 − 12.2197i −0.556579 − 1.51566i
\(66\) 0 0
\(67\) −0.637363 − 7.28509i −0.0778663 − 0.890016i −0.930209 − 0.367031i \(-0.880374\pi\)
0.852342 − 0.522984i \(-0.175181\pi\)
\(68\) 0 0
\(69\) −0.439018 + 0.760402i −0.0528516 + 0.0915416i
\(70\) 0 0
\(71\) 2.69808 − 0.475744i 0.320203 − 0.0564604i −0.0112367 − 0.999937i \(-0.503577\pi\)
0.331439 + 0.943476i \(0.392466\pi\)
\(72\) 0 0
\(73\) −1.74492 − 3.74199i −0.204227 − 0.437967i 0.777341 − 0.629080i \(-0.216568\pi\)
−0.981568 + 0.191113i \(0.938790\pi\)
\(74\) 0 0
\(75\) 0.855965 + 1.80849i 0.0988384 + 0.208827i
\(76\) 0 0
\(77\) −5.36285 + 5.36285i −0.611153 + 0.611153i
\(78\) 0 0
\(79\) −8.14836 − 2.96576i −0.916763 − 0.333674i −0.159813 − 0.987147i \(-0.551089\pi\)
−0.756950 + 0.653473i \(0.773311\pi\)
\(80\) 0 0
\(81\) −1.31702 − 7.46922i −0.146336 − 0.829913i
\(82\) 0 0
\(83\) −14.3797 + 3.85303i −1.57838 + 0.422925i −0.938423 − 0.345488i \(-0.887713\pi\)
−0.639954 + 0.768413i \(0.721047\pi\)
\(84\) 0 0
\(85\) 5.80192 + 6.87436i 0.629306 + 0.745629i
\(86\) 0 0
\(87\) −1.74055 − 0.466379i −0.186607 − 0.0500011i
\(88\) 0 0
\(89\) −1.72328 + 0.627223i −0.182667 + 0.0664855i −0.431734 − 0.902001i \(-0.642098\pi\)
0.249067 + 0.968486i \(0.419876\pi\)
\(90\) 0 0
\(91\) 22.3226 + 3.93607i 2.34004 + 0.412612i
\(92\) 0 0
\(93\) −1.37817 − 0.120574i −0.142909 − 0.0125029i
\(94\) 0 0
\(95\) 8.38177 − 4.97453i 0.859951 − 0.510376i
\(96\) 0 0
\(97\) −0.221785 − 0.0194037i −0.0225189 − 0.00197015i 0.0758907 − 0.997116i \(-0.475820\pi\)
−0.0984096 + 0.995146i \(0.531376\pi\)
\(98\) 0 0
\(99\) −5.44767 − 0.960571i −0.547511 − 0.0965410i
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.13.5 ✓ 120
5.2 odd 4 inner 380.2.bh.a.317.6 yes 120
19.3 odd 18 inner 380.2.bh.a.193.6 yes 120
95.22 even 36 inner 380.2.bh.a.117.5 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.13.5 ✓ 120 1.1 even 1 trivial
380.2.bh.a.117.5 yes 120 95.22 even 36 inner
380.2.bh.a.193.6 yes 120 19.3 odd 18 inner
380.2.bh.a.317.6 yes 120 5.2 odd 4 inner