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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(159,380)] 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("380.159"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 2])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.p (of order \(6\), degree \(2\), 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: \(10.3542500457\)
Analytic rank: \(0\)
Dimension: \(232\)
Relative dimension: \(116\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 159.9
Character \(\chi\) \(=\) 380.159
Dual form 380.3.p.a.239.9

$q$-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+(-1.95148 - 0.437846i) q^{2} +(2.80787 + 4.86337i) q^{3} +(3.61658 + 1.70890i) q^{4} +(4.08358 + 2.88520i) q^{5} +(-3.35010 - 10.7202i) q^{6} +7.39194 q^{7} +(-6.30947 - 4.91840i) q^{8} +(-11.2682 + 19.5171i) q^{9} +(-6.70576 - 7.41841i) q^{10} +15.7266i q^{11} +(1.84387 + 22.3871i) q^{12} +(-10.9304 - 6.31066i) q^{13} +(-14.4253 - 3.23653i) q^{14} +(-2.56567 + 27.9612i) q^{15} +(10.1593 + 12.3607i) q^{16} +(9.82197 - 5.67071i) q^{17} +(30.5353 - 33.1536i) q^{18} +(5.94597 - 18.0456i) q^{19} +(9.83806 + 17.4130i) q^{20} +(20.7556 + 35.9497i) q^{21} +(6.88583 - 30.6902i) q^{22} +(7.46080 - 12.9225i) q^{23} +(6.20383 - 44.4954i) q^{24} +(8.35120 + 23.5639i) q^{25} +(18.5674 + 17.1010i) q^{26} -76.0170 q^{27} +(26.7336 + 12.6321i) q^{28} +(5.52323 - 9.56651i) q^{29} +(17.2496 - 53.4425i) q^{30} -33.3774i q^{31} +(-14.4137 - 28.5700i) q^{32} +(-76.4843 + 44.1582i) q^{33} +(-21.6503 + 6.76580i) q^{34} +(30.1856 + 21.3273i) q^{35} +(-74.1053 + 51.3290i) q^{36} -5.93857i q^{37} +(-19.5047 + 32.6124i) q^{38} -70.8779i q^{39} +(-11.5746 - 38.2887i) q^{40} +(28.7657 + 49.8236i) q^{41} +(-24.7637 - 79.2430i) q^{42} +(-19.5535 - 33.8676i) q^{43} +(-26.8752 + 56.8766i) q^{44} +(-102.326 + 47.1886i) q^{45} +(-20.2177 + 21.9513i) q^{46} +(38.9526 - 67.4679i) q^{47} +(-31.5888 + 84.1158i) q^{48} +5.64076 q^{49} +(-5.97987 - 49.6411i) q^{50} +(55.1575 + 31.8452i) q^{51} +(-28.7463 - 41.5019i) q^{52} +(22.7610 + 13.1410i) q^{53} +(148.346 + 33.2838i) q^{54} +(-45.3745 + 64.2208i) q^{55} +(-46.6392 - 36.3565i) q^{56} +(104.458 - 21.7523i) q^{57} +(-14.9672 + 16.2506i) q^{58} +(-55.0984 + 31.8111i) q^{59} +(-57.0618 + 96.7395i) q^{60} +(9.38744 - 16.2595i) q^{61} +(-14.6142 + 65.1356i) q^{62} +(-83.2940 + 144.269i) q^{63} +(15.6187 + 62.0649i) q^{64} +(-26.4275 - 57.3064i) q^{65} +(168.592 - 52.6857i) q^{66} +(4.89670 - 8.48133i) q^{67} +(45.2126 - 3.72385i) q^{68} +83.7957 q^{69} +(-49.5686 - 54.8364i) q^{70} +(-45.6383 + 26.3493i) q^{71} +(167.089 - 67.7211i) q^{72} +(-88.9540 + 51.3576i) q^{73} +(-2.60018 + 11.5890i) q^{74} +(-91.1508 + 106.779i) q^{75} +(52.3423 - 55.1025i) q^{76} +116.250i q^{77} +(-31.0336 + 138.317i) q^{78} +(41.1038 - 23.7313i) q^{79} +(5.82310 + 79.7878i) q^{80} +(-112.032 - 194.044i) q^{81} +(-34.3207 - 109.825i) q^{82} +72.7504 q^{83} +(13.6298 + 165.484i) q^{84} +(56.4699 + 5.18158i) q^{85} +(23.3295 + 74.6535i) q^{86} +62.0339 q^{87} +(77.3497 - 99.2265i) q^{88} +(38.8438 - 67.2794i) q^{89} +(220.348 - 47.2849i) q^{90} +(-80.7967 - 46.6480i) q^{91} +(49.0658 - 33.9855i) q^{92} +(162.327 - 93.7194i) q^{93} +(-105.556 + 114.607i) q^{94} +(76.3462 - 56.5354i) q^{95} +(98.4749 - 150.320i) q^{96} +(-119.708 + 69.1136i) q^{97} +(-11.0079 - 2.46979i) q^{98} +(-306.938 - 177.211i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 232 q - 2 q^{5} + 8 q^{6} - 328 q^{9} + 20 q^{14} + 12 q^{16} + 92 q^{20} - 40 q^{21} - 134 q^{24} - 2 q^{25} + 28 q^{26} - 4 q^{29} + 268 q^{30} - 70 q^{34} + 12 q^{36} - 42 q^{40} - 12 q^{41} + 98 q^{44}+ \cdots - 628 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.95148 − 0.437846i −0.975742 − 0.218923i
\(3\) 2.80787 + 4.86337i 0.935955 + 1.62112i 0.772922 + 0.634502i \(0.218795\pi\)
0.163034 + 0.986621i \(0.447872\pi\)
\(4\) 3.61658 + 1.70890i 0.904145 + 0.427225i
\(5\) 4.08358 + 2.88520i 0.816715 + 0.577041i
\(6\) −3.35010 − 10.7202i −0.558350 − 1.78670i
\(7\) 7.39194 1.05599 0.527996 − 0.849247i \(-0.322944\pi\)
0.527996 + 0.849247i \(0.322944\pi\)
\(8\) −6.30947 − 4.91840i −0.788683 − 0.614800i
\(9\) −11.2682 + 19.5171i −1.25202 + 2.16857i
\(10\) −6.70576 − 7.41841i −0.670576 − 0.741841i
\(11\) 15.7266i 1.42969i 0.699282 + 0.714846i \(0.253503\pi\)
−0.699282 + 0.714846i \(0.746497\pi\)
\(12\) 1.84387 + 22.3871i 0.153656 + 1.86559i
\(13\) −10.9304 − 6.31066i −0.840799 − 0.485435i 0.0167370 − 0.999860i \(-0.494672\pi\)
−0.857536 + 0.514425i \(0.828006\pi\)
\(14\) −14.4253 − 3.23653i −1.03038 − 0.231181i
\(15\) −2.56567 + 27.9612i −0.171045 + 1.86408i
\(16\) 10.1593 + 12.3607i 0.634958 + 0.772547i
\(17\) 9.82197 − 5.67071i 0.577763 − 0.333571i −0.182481 − 0.983209i \(-0.558413\pi\)
0.760244 + 0.649638i \(0.225079\pi\)
\(18\) 30.5353 − 33.1536i 1.69640 − 1.84187i
\(19\) 5.94597 − 18.0456i 0.312946 − 0.949771i
\(20\) 9.83806 + 17.4130i 0.491903 + 0.870650i
\(21\) 20.7556 + 35.9497i 0.988361 + 1.71189i
\(22\) 6.88583 − 30.6902i 0.312992 − 1.39501i
\(23\) 7.46080 − 12.9225i 0.324383 − 0.561847i −0.657005 − 0.753887i \(-0.728177\pi\)
0.981387 + 0.192039i \(0.0615101\pi\)
\(24\) 6.20383 − 44.4954i 0.258493 − 1.85398i
\(25\) 8.35120 + 23.5639i 0.334048 + 0.942556i
\(26\) 18.5674 + 17.1010i 0.714130 + 0.657730i
\(27\) −76.0170 −2.81545
\(28\) 26.7336 + 12.6321i 0.954770 + 0.451146i
\(29\) 5.52323 − 9.56651i 0.190456 − 0.329880i −0.754945 − 0.655788i \(-0.772337\pi\)
0.945401 + 0.325908i \(0.105670\pi\)
\(30\) 17.2496 − 53.4425i 0.574985 − 1.78142i
\(31\) − 33.3774i − 1.07669i −0.842724 − 0.538346i \(-0.819049\pi\)
0.842724 − 0.538346i \(-0.180951\pi\)
\(32\) −14.4137 − 28.5700i −0.450427 − 0.892813i
\(33\) −76.4843 + 44.1582i −2.31770 + 1.33813i
\(34\) −21.6503 + 6.76580i −0.636774 + 0.198994i
\(35\) 30.1856 + 21.3273i 0.862444 + 0.609350i
\(36\) −74.1053 + 51.3290i −2.05848 + 1.42581i
\(37\) − 5.93857i − 0.160502i −0.996775 − 0.0802509i \(-0.974428\pi\)
0.996775 − 0.0802509i \(-0.0255722\pi\)
\(38\) −19.5047 + 32.6124i −0.513281 + 0.858220i
\(39\) − 70.8779i − 1.81738i
\(40\) −11.5746 − 38.2887i −0.289365 − 0.957219i
\(41\) 28.7657 + 49.8236i 0.701602 + 1.21521i 0.967904 + 0.251321i \(0.0808649\pi\)
−0.266302 + 0.963890i \(0.585802\pi\)
\(42\) −24.7637 − 79.2430i −0.589613 − 1.88674i
\(43\) −19.5535 − 33.8676i −0.454732 − 0.787618i 0.543941 − 0.839123i \(-0.316932\pi\)
−0.998673 + 0.0515050i \(0.983598\pi\)
\(44\) −26.8752 + 56.8766i −0.610800 + 1.29265i
\(45\) −102.326 + 47.1886i −2.27390 + 1.04864i
\(46\) −20.2177 + 21.9513i −0.439515 + 0.477203i
\(47\) 38.9526 − 67.4679i 0.828779 − 1.43549i −0.0702177 − 0.997532i \(-0.522369\pi\)
0.898997 − 0.437956i \(-0.144297\pi\)
\(48\) −31.5888 + 84.1158i −0.658101 + 1.75241i
\(49\) 5.64076 0.115118
\(50\) −5.97987 − 49.6411i −0.119597 − 0.992822i
\(51\) 55.1575 + 31.8452i 1.08152 + 0.624416i
\(52\) −28.7463 − 41.5019i −0.552814 − 0.798114i
\(53\) 22.7610 + 13.1410i 0.429452 + 0.247944i 0.699113 − 0.715011i \(-0.253578\pi\)
−0.269661 + 0.962955i \(0.586912\pi\)
\(54\) 148.346 + 33.2838i 2.74715 + 0.616366i
\(55\) −45.3745 + 64.2208i −0.824990 + 1.16765i
\(56\) −46.6392 − 36.3565i −0.832843 − 0.649223i
\(57\) 104.458 − 21.7523i 1.83260 − 0.381620i
\(58\) −14.9672 + 16.2506i −0.258054 + 0.280182i
\(59\) −55.0984 + 31.8111i −0.933870 + 0.539170i −0.888034 − 0.459779i \(-0.847929\pi\)
−0.0458369 + 0.998949i \(0.514595\pi\)
\(60\) −57.0618 + 96.7395i −0.951030 + 1.61232i
\(61\) 9.38744 − 16.2595i 0.153892 − 0.266550i −0.778763 − 0.627319i \(-0.784152\pi\)
0.932655 + 0.360769i \(0.117486\pi\)
\(62\) −14.6142 + 65.1356i −0.235713 + 1.05057i
\(63\) −83.2940 + 144.269i −1.32213 + 2.28999i
\(64\) 15.6187 + 62.0649i 0.244043 + 0.969764i
\(65\) −26.4275 − 57.3064i −0.406577 − 0.881638i
\(66\) 168.592 − 52.6857i 2.55443 − 0.798268i
\(67\) 4.89670 − 8.48133i 0.0730851 − 0.126587i −0.827167 − 0.561956i \(-0.810049\pi\)
0.900252 + 0.435369i \(0.143382\pi\)
\(68\) 45.2126 − 3.72385i 0.664891 − 0.0547625i
\(69\) 83.7957 1.21443
\(70\) −49.5686 − 54.8364i −0.708122 − 0.783377i
\(71\) −45.6383 + 26.3493i −0.642793 + 0.371117i −0.785690 − 0.618621i \(-0.787692\pi\)
0.142896 + 0.989738i \(0.454358\pi\)
\(72\) 167.089 − 67.7211i 2.32069 − 0.940571i
\(73\) −88.9540 + 51.3576i −1.21855 + 0.703529i −0.964607 − 0.263691i \(-0.915060\pi\)
−0.253940 + 0.967220i \(0.581727\pi\)
\(74\) −2.60018 + 11.5890i −0.0351375 + 0.156608i
\(75\) −91.1508 + 106.779i −1.21534 + 1.42372i
\(76\) 52.3423 − 55.1025i 0.688714 − 0.725033i
\(77\) 116.250i 1.50974i
\(78\) −31.0336 + 138.317i −0.397867 + 1.77330i
\(79\) 41.1038 − 23.7313i 0.520301 − 0.300396i −0.216757 − 0.976226i \(-0.569548\pi\)
0.737058 + 0.675829i \(0.236214\pi\)
\(80\) 5.82310 + 79.7878i 0.0727887 + 0.997347i
\(81\) −112.032 − 194.044i −1.38311 − 2.39561i
\(82\) −34.3207 − 109.825i −0.418545 − 1.33933i
\(83\) 72.7504 0.876511 0.438256 − 0.898850i \(-0.355596\pi\)
0.438256 + 0.898850i \(0.355596\pi\)
\(84\) 13.6298 + 165.484i 0.162259 + 1.97005i
\(85\) 56.4699 + 5.18158i 0.664352 + 0.0609597i
\(86\) 23.3295 + 74.6535i 0.271273 + 0.868064i
\(87\) 62.0339 0.713034
\(88\) 77.3497 − 99.2265i 0.878974 − 1.12757i
\(89\) 38.8438 − 67.2794i 0.436447 − 0.755949i −0.560965 − 0.827839i \(-0.689570\pi\)
0.997413 + 0.0718906i \(0.0229033\pi\)
\(90\) 220.348 − 47.2849i 2.44831 − 0.525388i
\(91\) −80.7967 − 46.6480i −0.887876 − 0.512615i
\(92\) 49.0658 − 33.9855i 0.533324 − 0.369407i
\(93\) 162.327 − 93.7194i 1.74545 − 1.00774i
\(94\) −105.556 + 114.607i −1.12294 + 1.21923i
\(95\) 76.3462 − 56.5354i 0.803644 − 0.595110i
\(96\) 98.4749 − 150.320i 1.02578 − 1.56583i
\(97\) −119.708 + 69.1136i −1.23411 + 0.712511i −0.967883 − 0.251401i \(-0.919109\pi\)
−0.266222 + 0.963912i \(0.585776\pi\)
\(98\) −11.0079 − 2.46979i −0.112325 − 0.0252019i
\(99\) −306.938 − 177.211i −3.10039 − 1.79001i
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 380.3.p.a.159.9 ✓ 232
4.3 odd 2 inner 380.3.p.a.159.47 yes 232
5.4 even 2 inner 380.3.p.a.159.108 yes 232
19.11 even 3 inner 380.3.p.a.239.70 yes 232
20.19 odd 2 inner 380.3.p.a.159.70 yes 232
76.11 odd 6 inner 380.3.p.a.239.108 yes 232
95.49 even 6 inner 380.3.p.a.239.47 yes 232
380.239 odd 6 inner 380.3.p.a.239.9 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.9 ✓ 232 1.1 even 1 trivial
380.3.p.a.159.47 yes 232 4.3 odd 2 inner
380.3.p.a.159.70 yes 232 20.19 odd 2 inner
380.3.p.a.159.108 yes 232 5.4 even 2 inner
380.3.p.a.239.9 yes 232 380.239 odd 6 inner
380.3.p.a.239.47 yes 232 95.49 even 6 inner
380.3.p.a.239.70 yes 232 19.11 even 3 inner
380.3.p.a.239.108 yes 232 76.11 odd 6 inner