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Properties

Label 380.2.bd.a.149.1
Level $380$
Weight $2$
Character 380.149
Analytic conductor $3.034$
Analytic rank $0$
Dimension $60$
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(9,380)] 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("380.9"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 9, 8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.bd (of order \(18\), degree \(6\), 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: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 149.1
Character \(\chi\) \(=\) 380.149
Dual form 380.2.bd.a.329.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00692 - 2.76648i) q^{3} +(-1.18175 - 1.89828i) q^{5} +(4.39465 - 2.53725i) q^{7} +(-4.34139 + 3.64286i) q^{9} +(-0.0455037 + 0.0788147i) q^{11} +(1.49393 - 4.10453i) q^{13} +(-4.06164 + 5.18068i) q^{15} +(-2.60202 + 3.10096i) q^{17} +(2.14085 + 3.79694i) q^{19} +(-11.4443 - 9.60291i) q^{21} +(-5.18504 + 0.914263i) q^{23} +(-2.20696 + 4.48657i) q^{25} +(6.80050 + 3.92627i) q^{27} +(2.94175 - 2.46842i) q^{29} +(1.17517 + 2.03546i) q^{31} +(0.263858 + 0.0465252i) q^{33} +(-10.0098 - 5.34391i) q^{35} -3.24116i q^{37} -12.8593 q^{39} +(0.915043 - 0.333048i) q^{41} +(0.609246 + 0.107427i) q^{43} +(12.0456 + 3.93626i) q^{45} +(6.04820 + 7.20797i) q^{47} +(9.37531 - 16.2385i) q^{49} +(11.1988 + 4.07601i) q^{51} +(0.0924433 - 0.0163003i) q^{53} +(0.203386 - 0.00676001i) q^{55} +(8.34850 - 9.74582i) q^{57} +(9.83226 + 8.25025i) q^{59} +(-1.27191 - 7.21335i) q^{61} +(-9.83604 + 27.0243i) q^{63} +(-9.55699 + 2.01461i) q^{65} +(-4.46021 - 5.31547i) q^{67} +(7.75019 + 13.4237i) q^{69} +(2.28841 - 12.9782i) q^{71} +(-1.36598 - 3.75301i) q^{73} +(14.6342 + 1.58790i) q^{75} +0.461818i q^{77} +(1.15586 - 0.420699i) q^{79} +(1.06207 - 6.02331i) q^{81} +(3.60686 - 2.08242i) q^{83} +(8.96143 + 1.27482i) q^{85} +(-9.79091 - 5.65279i) q^{87} +(-3.16508 - 1.15200i) q^{89} +(-3.84894 - 21.8284i) q^{91} +(4.44775 - 5.30063i) q^{93} +(4.67773 - 8.55096i) q^{95} +(-8.62996 + 10.2848i) q^{97} +(-0.0895616 - 0.507929i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q - 6 q^{11} + 33 q^{15} + 24 q^{19} - 24 q^{21} - 6 q^{25} + 12 q^{29} - 21 q^{35} - 24 q^{39} - 6 q^{41} - 42 q^{45} + 60 q^{49} + 36 q^{51} - 45 q^{55} + 60 q^{61} - 15 q^{65} - 48 q^{69} + 36 q^{71}+ \cdots - 162 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{2}{9}\right)\) \(-1\) \(1\)

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
\(3\) −1.00692 − 2.76648i −0.581343 − 1.59723i −0.785887 − 0.618370i \(-0.787793\pi\)
0.204544 − 0.978857i \(-0.434429\pi\)
\(4\) 0 0
\(5\) −1.18175 − 1.89828i −0.528492 − 0.848938i
\(6\) 0 0
\(7\) 4.39465 − 2.53725i 1.66102 − 0.958992i 0.688794 − 0.724957i \(-0.258140\pi\)
0.972228 − 0.234035i \(-0.0751929\pi\)
\(8\) 0 0
\(9\) −4.34139 + 3.64286i −1.44713 + 1.21429i
\(10\) 0 0
\(11\) −0.0455037 + 0.0788147i −0.0137199 + 0.0237635i −0.872804 − 0.488071i \(-0.837701\pi\)
0.859084 + 0.511835i \(0.171034\pi\)
\(12\) 0 0
\(13\) 1.49393 − 4.10453i 0.414340 − 1.13839i −0.540519 − 0.841332i \(-0.681772\pi\)
0.954859 − 0.297059i \(-0.0960058\pi\)
\(14\) 0 0
\(15\) −4.06164 + 5.18068i −1.04871 + 1.33765i
\(16\) 0 0
\(17\) −2.60202 + 3.10096i −0.631082 + 0.752094i −0.982933 − 0.183962i \(-0.941108\pi\)
0.351852 + 0.936056i \(0.385552\pi\)
\(18\) 0 0
\(19\) 2.14085 + 3.79694i 0.491145 + 0.871078i
\(20\) 0 0
\(21\) −11.4443 − 9.60291i −2.49735 − 2.09553i
\(22\) 0 0
\(23\) −5.18504 + 0.914263i −1.08116 + 0.190637i −0.685726 − 0.727859i \(-0.740515\pi\)
−0.395429 + 0.918496i \(0.629404\pi\)
\(24\) 0 0
\(25\) −2.20696 + 4.48657i −0.441391 + 0.897315i
\(26\) 0 0
\(27\) 6.80050 + 3.92627i 1.30876 + 0.755611i
\(28\) 0 0
\(29\) 2.94175 − 2.46842i 0.546269 − 0.458374i −0.327407 − 0.944884i \(-0.606175\pi\)
0.873675 + 0.486510i \(0.161730\pi\)
\(30\) 0 0
\(31\) 1.17517 + 2.03546i 0.211067 + 0.365579i 0.952049 − 0.305946i \(-0.0989728\pi\)
−0.740982 + 0.671525i \(0.765639\pi\)
\(32\) 0 0
\(33\) 0.263858 + 0.0465252i 0.0459317 + 0.00809900i
\(34\) 0 0
\(35\) −10.0098 − 5.34391i −1.69196 − 0.903285i
\(36\) 0 0
\(37\) − 3.24116i − 0.532843i −0.963857 − 0.266421i \(-0.914159\pi\)
0.963857 − 0.266421i \(-0.0858413\pi\)
\(38\) 0 0
\(39\) −12.8593 −2.05914
\(40\) 0 0
\(41\) 0.915043 − 0.333048i 0.142906 − 0.0520134i −0.269577 − 0.962979i \(-0.586884\pi\)
0.412483 + 0.910965i \(0.364662\pi\)
\(42\) 0 0
\(43\) 0.609246 + 0.107427i 0.0929092 + 0.0163824i 0.219909 − 0.975520i \(-0.429424\pi\)
−0.127000 + 0.991903i \(0.540535\pi\)
\(44\) 0 0
\(45\) 12.0456 + 3.93626i 1.79565 + 0.586782i
\(46\) 0 0
\(47\) 6.04820 + 7.20797i 0.882221 + 1.05139i 0.998308 + 0.0581509i \(0.0185204\pi\)
−0.116087 + 0.993239i \(0.537035\pi\)
\(48\) 0 0
\(49\) 9.37531 − 16.2385i 1.33933 − 2.31979i
\(50\) 0 0
\(51\) 11.1988 + 4.07601i 1.56814 + 0.570756i
\(52\) 0 0
\(53\) 0.0924433 − 0.0163003i 0.0126981 − 0.00223901i −0.167296 − 0.985907i \(-0.553503\pi\)
0.179994 + 0.983668i \(0.442392\pi\)
\(54\) 0 0
\(55\) 0.203386 − 0.00676001i 0.0274246 − 0.000911520i
\(56\) 0 0
\(57\) 8.34850 − 9.74582i 1.10579 − 1.29087i
\(58\) 0 0
\(59\) 9.83226 + 8.25025i 1.28005 + 1.07409i 0.993237 + 0.116105i \(0.0370409\pi\)
0.286815 + 0.957986i \(0.407404\pi\)
\(60\) 0 0
\(61\) −1.27191 − 7.21335i −0.162851 − 0.923574i −0.951253 − 0.308413i \(-0.900202\pi\)
0.788401 − 0.615161i \(-0.210909\pi\)
\(62\) 0 0
\(63\) −9.83604 + 27.0243i −1.23922 + 3.40474i
\(64\) 0 0
\(65\) −9.55699 + 2.01461i −1.18540 + 0.249882i
\(66\) 0 0
\(67\) −4.46021 − 5.31547i −0.544901 − 0.649388i 0.421378 − 0.906885i \(-0.361547\pi\)
−0.966279 + 0.257497i \(0.917102\pi\)
\(68\) 0 0
\(69\) 7.75019 + 13.4237i 0.933013 + 1.61603i
\(70\) 0 0
\(71\) 2.28841 − 12.9782i 0.271584 − 1.54023i −0.478024 − 0.878347i \(-0.658647\pi\)
0.749608 − 0.661882i \(-0.230242\pi\)
\(72\) 0 0
\(73\) −1.36598 − 3.75301i −0.159876 − 0.439257i 0.833728 − 0.552176i \(-0.186202\pi\)
−0.993604 + 0.112919i \(0.963980\pi\)
\(74\) 0 0
\(75\) 14.6342 + 1.58790i 1.68982 + 0.183354i
\(76\) 0 0
\(77\) 0.461818i 0.0526290i
\(78\) 0 0
\(79\) 1.15586 − 0.420699i 0.130045 − 0.0473323i −0.276178 − 0.961106i \(-0.589068\pi\)
0.406223 + 0.913774i \(0.366846\pi\)
\(80\) 0 0
\(81\) 1.06207 − 6.02331i 0.118008 − 0.669257i
\(82\) 0 0
\(83\) 3.60686 − 2.08242i 0.395905 − 0.228576i −0.288811 − 0.957386i \(-0.593260\pi\)
0.684715 + 0.728811i \(0.259927\pi\)
\(84\) 0 0
\(85\) 8.96143 + 1.27482i 0.972003 + 0.138273i
\(86\) 0 0
\(87\) −9.79091 − 5.65279i −1.04970 − 0.606042i
\(88\) 0 0
\(89\) −3.16508 − 1.15200i −0.335498 − 0.122111i 0.168777 − 0.985654i \(-0.446018\pi\)
−0.504275 + 0.863543i \(0.668240\pi\)
\(90\) 0 0
\(91\) −3.84894 − 21.8284i −0.403479 − 2.28824i
\(92\) 0 0
\(93\) 4.44775 − 5.30063i 0.461210 − 0.549649i
\(94\) 0 0
\(95\) 4.67773 − 8.55096i 0.479925 − 0.877310i
\(96\) 0 0
\(97\) −8.62996 + 10.2848i −0.876239 + 1.04426i 0.122419 + 0.992478i \(0.460935\pi\)
−0.998658 + 0.0517827i \(0.983510\pi\)
\(98\) 0 0
\(99\) −0.0895616 − 0.507929i −0.00900128 − 0.0510488i
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.bd.a.149.1 ✓ 60
5.4 even 2 inner 380.2.bd.a.149.10 yes 60
19.6 even 9 inner 380.2.bd.a.329.10 yes 60
95.44 even 18 inner 380.2.bd.a.329.1 yes 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bd.a.149.1 ✓ 60 1.1 even 1 trivial
380.2.bd.a.149.10 yes 60 5.4 even 2 inner
380.2.bd.a.329.1 yes 60 95.44 even 18 inner
380.2.bd.a.329.10 yes 60 19.6 even 9 inner