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Properties

Label 380.2.bh.a.33.7
Level $380$
Weight $2$
Character 380.33
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 33.7
Character \(\chi\) \(=\) 380.33
Dual form 380.2.bh.a.357.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.377219 + 0.538725i) q^{3} +(2.02714 + 0.943776i) q^{5} +(1.97160 + 0.528288i) q^{7} +(0.878130 - 2.41264i) q^{9} +(-0.906926 + 1.57084i) q^{11} +(-1.44805 - 1.01394i) q^{13} +(0.256239 + 1.44808i) q^{15} +(0.0577770 - 0.123903i) q^{17} +(-2.93922 + 3.21885i) q^{19} +(0.459122 + 1.26143i) q^{21} +(6.07446 + 0.531447i) q^{23} +(3.21857 + 3.82633i) q^{25} +(3.53675 - 0.947670i) q^{27} +(-3.14393 - 1.14430i) q^{29} +(0.680748 - 0.393030i) q^{31} +(-1.18836 + 0.103968i) q^{33} +(3.49811 + 2.93166i) q^{35} +(-4.83846 - 4.83846i) q^{37} -1.16258i q^{39} +(5.31428 + 0.937051i) q^{41} +(0.277984 + 3.17737i) q^{43} +(4.05709 - 4.06200i) q^{45} +(-5.00993 + 2.33617i) q^{47} +(-2.45407 - 1.41686i) q^{49} +(0.0885442 - 0.0156127i) q^{51} +(0.933570 - 10.6708i) q^{53} +(-3.32099 + 2.32838i) q^{55} +(-2.84280 - 0.369220i) q^{57} +(-8.38425 + 3.05162i) q^{59} +(-5.98020 + 5.01798i) q^{61} +(3.00589 - 4.29286i) q^{63} +(-1.97847 - 3.42202i) q^{65} +(-3.83106 - 8.21573i) q^{67} +(2.00510 + 3.47293i) q^{69} +(-1.87804 + 2.23816i) q^{71} +(2.70247 - 1.89229i) q^{73} +(-0.847230 + 3.17729i) q^{75} +(-2.61795 + 2.61795i) q^{77} +(-2.28065 + 12.9342i) q^{79} +(-4.05575 - 3.40318i) q^{81} +(4.13317 - 15.4252i) q^{83} +(0.234059 - 0.196640i) q^{85} +(-0.569489 - 2.12536i) q^{87} +(-1.72877 - 9.80435i) q^{89} +(-2.31932 - 2.76406i) q^{91} +(0.468526 + 0.218477i) q^{93} +(-8.99608 + 3.75108i) q^{95} +(-5.79625 - 2.70284i) q^{97} +(2.99348 + 3.56749i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 6 q^{7} + 18 q^{15} - 18 q^{17} + 48 q^{21} - 36 q^{23} - 24 q^{25} - 60 q^{33} - 18 q^{35} - 12 q^{41} - 36 q^{43} + 18 q^{45} - 24 q^{47} + 96 q^{51} - 18 q^{53} + 72 q^{55} - 6 q^{57} - 24 q^{61}+ \cdots - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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
\(3\) 0.377219 + 0.538725i 0.217787 + 0.311033i 0.913148 − 0.407628i \(-0.133644\pi\)
−0.695360 + 0.718661i \(0.744755\pi\)
\(4\) 0 0
\(5\) 2.02714 + 0.943776i 0.906563 + 0.422070i
\(6\) 0 0
\(7\) 1.97160 + 0.528288i 0.745194 + 0.199674i 0.611385 − 0.791333i \(-0.290613\pi\)
0.133809 + 0.991007i \(0.457279\pi\)
\(8\) 0 0
\(9\) 0.878130 − 2.41264i 0.292710 − 0.804215i
\(10\) 0 0
\(11\) −0.906926 + 1.57084i −0.273448 + 0.473626i −0.969742 − 0.244130i \(-0.921498\pi\)
0.696294 + 0.717757i \(0.254831\pi\)
\(12\) 0 0
\(13\) −1.44805 − 1.01394i −0.401617 − 0.281215i 0.355267 − 0.934765i \(-0.384390\pi\)
−0.756883 + 0.653550i \(0.773279\pi\)
\(14\) 0 0
\(15\) 0.256239 + 1.44808i 0.0661607 + 0.373892i
\(16\) 0 0
\(17\) 0.0577770 − 0.123903i 0.0140130 − 0.0300509i −0.899177 − 0.437586i \(-0.855834\pi\)
0.913190 + 0.407535i \(0.133611\pi\)
\(18\) 0 0
\(19\) −2.93922 + 3.21885i −0.674304 + 0.738454i
\(20\) 0 0
\(21\) 0.459122 + 1.26143i 0.100189 + 0.275266i
\(22\) 0 0
\(23\) 6.07446 + 0.531447i 1.26661 + 0.110814i 0.700583 − 0.713571i \(-0.252923\pi\)
0.566030 + 0.824385i \(0.308479\pi\)
\(24\) 0 0
\(25\) 3.21857 + 3.82633i 0.643715 + 0.765266i
\(26\) 0 0
\(27\) 3.53675 − 0.947670i 0.680649 − 0.182379i
\(28\) 0 0
\(29\) −3.14393 − 1.14430i −0.583813 − 0.212491i 0.0331929 − 0.999449i \(-0.489432\pi\)
−0.617006 + 0.786958i \(0.711655\pi\)
\(30\) 0 0
\(31\) 0.680748 − 0.393030i 0.122266 − 0.0705903i −0.437620 − 0.899160i \(-0.644178\pi\)
0.559886 + 0.828570i \(0.310845\pi\)
\(32\) 0 0
\(33\) −1.18836 + 0.103968i −0.206867 + 0.0180985i
\(34\) 0 0
\(35\) 3.49811 + 2.93166i 0.591289 + 0.495541i
\(36\) 0 0
\(37\) −4.83846 − 4.83846i −0.795439 − 0.795439i 0.186934 − 0.982372i \(-0.440145\pi\)
−0.982372 + 0.186934i \(0.940145\pi\)
\(38\) 0 0
\(39\) − 1.16258i − 0.186161i
\(40\) 0 0
\(41\) 5.31428 + 0.937051i 0.829951 + 0.146343i 0.572455 − 0.819936i \(-0.305991\pi\)
0.257497 + 0.966279i \(0.417102\pi\)
\(42\) 0 0
\(43\) 0.277984 + 3.17737i 0.0423922 + 0.484545i 0.987482 + 0.157733i \(0.0504186\pi\)
−0.945090 + 0.326811i \(0.894026\pi\)
\(44\) 0 0
\(45\) 4.05709 − 4.06200i 0.604795 − 0.605527i
\(46\) 0 0
\(47\) −5.00993 + 2.33617i −0.730773 + 0.340765i −0.752137 − 0.659007i \(-0.770977\pi\)
0.0213641 + 0.999772i \(0.493199\pi\)
\(48\) 0 0
\(49\) −2.45407 − 1.41686i −0.350581 − 0.202408i
\(50\) 0 0
\(51\) 0.0885442 − 0.0156127i 0.0123987 − 0.00218622i
\(52\) 0 0
\(53\) 0.933570 − 10.6708i 0.128236 − 1.46574i −0.610645 − 0.791905i \(-0.709090\pi\)
0.738881 − 0.673836i \(-0.235355\pi\)
\(54\) 0 0
\(55\) −3.32099 + 2.32838i −0.447802 + 0.313958i
\(56\) 0 0
\(57\) −2.84280 − 0.369220i −0.376538 − 0.0489044i
\(58\) 0 0
\(59\) −8.38425 + 3.05162i −1.09154 + 0.397287i −0.824190 − 0.566314i \(-0.808369\pi\)
−0.267346 + 0.963601i \(0.586147\pi\)
\(60\) 0 0
\(61\) −5.98020 + 5.01798i −0.765686 + 0.642487i −0.939600 − 0.342274i \(-0.888803\pi\)
0.173914 + 0.984761i \(0.444359\pi\)
\(62\) 0 0
\(63\) 3.00589 − 4.29286i 0.378707 − 0.540849i
\(64\) 0 0
\(65\) −1.97847 − 3.42202i −0.245399 − 0.424449i
\(66\) 0 0
\(67\) −3.83106 − 8.21573i −0.468038 − 1.00371i −0.988694 − 0.149944i \(-0.952091\pi\)
0.520657 − 0.853766i \(-0.325687\pi\)
\(68\) 0 0
\(69\) 2.00510 + 3.47293i 0.241386 + 0.418092i
\(70\) 0 0
\(71\) −1.87804 + 2.23816i −0.222883 + 0.265621i −0.865885 − 0.500243i \(-0.833244\pi\)
0.643002 + 0.765864i \(0.277688\pi\)
\(72\) 0 0
\(73\) 2.70247 − 1.89229i 0.316300 − 0.221476i −0.404635 − 0.914478i \(-0.632601\pi\)
0.720935 + 0.693003i \(0.243713\pi\)
\(74\) 0 0
\(75\) −0.847230 + 3.17729i −0.0978297 + 0.366882i
\(76\) 0 0
\(77\) −2.61795 + 2.61795i −0.298343 + 0.298343i
\(78\) 0 0
\(79\) −2.28065 + 12.9342i −0.256593 + 1.45521i 0.535357 + 0.844626i \(0.320177\pi\)
−0.791950 + 0.610586i \(0.790934\pi\)
\(80\) 0 0
\(81\) −4.05575 − 3.40318i −0.450639 − 0.378131i
\(82\) 0 0
\(83\) 4.13317 − 15.4252i 0.453674 − 1.69313i −0.238284 − 0.971196i \(-0.576585\pi\)
0.691957 − 0.721938i \(-0.256749\pi\)
\(84\) 0 0
\(85\) 0.234059 − 0.196640i 0.0253872 − 0.0213286i
\(86\) 0 0
\(87\) −0.569489 − 2.12536i −0.0610557 − 0.227863i
\(88\) 0 0
\(89\) −1.72877 − 9.80435i −0.183249 − 1.03926i −0.928184 − 0.372121i \(-0.878631\pi\)
0.744935 − 0.667137i \(-0.232481\pi\)
\(90\) 0 0
\(91\) −2.31932 − 2.76406i −0.243131 − 0.289752i
\(92\) 0 0
\(93\) 0.468526 + 0.218477i 0.0485839 + 0.0226550i
\(94\) 0 0
\(95\) −8.99608 + 3.75108i −0.922978 + 0.384853i
\(96\) 0 0
\(97\) −5.79625 − 2.70284i −0.588520 − 0.274431i 0.105462 − 0.994423i \(-0.466368\pi\)
−0.693982 + 0.719992i \(0.744145\pi\)
\(98\) 0 0
\(99\) 2.99348 + 3.56749i 0.300856 + 0.358546i
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.33.7 ✓ 120
5.2 odd 4 inner 380.2.bh.a.337.7 yes 120
19.15 odd 18 inner 380.2.bh.a.53.7 yes 120
95.72 even 36 inner 380.2.bh.a.357.7 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.33.7 ✓ 120 1.1 even 1 trivial
380.2.bh.a.53.7 yes 120 19.15 odd 18 inner
380.2.bh.a.337.7 yes 120 5.2 odd 4 inner
380.2.bh.a.357.7 yes 120 95.72 even 36 inner