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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 239.87
Character \(\chi\) \(=\) 380.239
Dual form 380.3.p.a.159.87

$q$-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+(1.42473 + 1.40361i) q^{2} +(-1.00965 + 1.74877i) q^{3} +(0.0597334 + 3.99955i) q^{4} +(3.69520 + 3.36831i) q^{5} +(-3.89309 + 1.07437i) q^{6} +3.07442 q^{7} +(-5.52873 + 5.78214i) q^{8} +(2.46120 + 4.26292i) q^{9} +(0.536862 + 9.98558i) q^{10} +3.77070i q^{11} +(-7.05462 - 3.93371i) q^{12} +(9.11726 - 5.26385i) q^{13} +(4.38023 + 4.31530i) q^{14} +(-9.62128 + 3.06123i) q^{15} +(-15.9929 + 0.477814i) q^{16} +(-9.93069 - 5.73349i) q^{17} +(-2.47694 + 9.52810i) q^{18} +(16.4485 - 9.51032i) q^{19} +(-13.2510 + 14.9803i) q^{20} +(-3.10410 + 5.37645i) q^{21} +(-5.29261 + 5.37225i) q^{22} +(-6.96607 - 12.0656i) q^{23} +(-4.52955 - 15.5064i) q^{24} +(2.30896 + 24.8931i) q^{25} +(20.3781 + 5.29753i) q^{26} -28.1136 q^{27} +(0.183645 + 12.2963i) q^{28} +(-9.09921 - 15.7603i) q^{29} +(-18.0045 - 9.14313i) q^{30} +50.1003i q^{31} +(-23.4562 - 21.7671i) q^{32} +(-6.59410 - 3.80710i) q^{33} +(-6.10099 - 22.1076i) q^{34} +(11.3606 + 10.3556i) q^{35} +(-16.9028 + 10.0983i) q^{36} -17.0262i q^{37} +(36.7836 + 9.53770i) q^{38} +21.2587i q^{39} +(-39.9058 + 2.74368i) q^{40} +(35.7872 - 61.9853i) q^{41} +(-11.9690 + 3.30306i) q^{42} +(-2.56904 + 4.44971i) q^{43} +(-15.0811 + 0.225237i) q^{44} +(-5.26423 + 24.0424i) q^{45} +(7.01063 - 26.9679i) q^{46} +(-27.8127 - 48.1730i) q^{47} +(15.3117 - 28.4503i) q^{48} -39.5480 q^{49} +(-31.6507 + 38.7070i) q^{50} +(20.0531 - 11.5777i) q^{51} +(21.5977 + 36.1506i) q^{52} +(-21.0636 + 12.1611i) q^{53} +(-40.0544 - 39.4607i) q^{54} +(-12.7009 + 13.9335i) q^{55} +(-16.9976 + 17.7767i) q^{56} +(0.0240704 + 38.3668i) q^{57} +(9.15743 - 35.2260i) q^{58} +(79.5696 + 45.9396i) q^{59} +(-12.8183 - 38.2980i) q^{60} +(-2.51799 - 4.36129i) q^{61} +(-70.3214 + 71.3795i) q^{62} +(7.56675 + 13.1060i) q^{63} +(-2.86635 - 63.9358i) q^{64} +(51.4204 + 11.2588i) q^{65} +(-4.05113 - 14.6797i) q^{66} +(-16.9406 - 29.3419i) q^{67} +(22.3382 - 40.0608i) q^{68} +28.1333 q^{69} +(1.65054 + 30.6998i) q^{70} +(112.787 + 65.1173i) q^{71} +(-38.2561 - 9.33752i) q^{72} +(69.2413 + 39.9765i) q^{73} +(23.8982 - 24.2578i) q^{74} +(-45.8637 - 21.0956i) q^{75} +(39.0196 + 65.2187i) q^{76} +11.5927i q^{77} +(-29.8390 + 30.2880i) q^{78} +(6.25869 + 3.61346i) q^{79} +(-60.7062 - 52.1033i) q^{80} +(6.23423 - 10.7980i) q^{81} +(137.991 - 38.0811i) q^{82} +61.2186 q^{83} +(-21.6888 - 12.0939i) q^{84} +(-17.3837 - 54.6360i) q^{85} +(-9.90588 + 2.73371i) q^{86} +36.7482 q^{87} +(-21.8027 - 20.8472i) q^{88} +(14.2293 + 24.6458i) q^{89} +(-41.2464 + 26.8651i) q^{90} +(28.0303 - 16.1833i) q^{91} +(47.8408 - 28.5819i) q^{92} +(-87.6139 - 50.5839i) q^{93} +(27.9906 - 107.672i) q^{94} +(92.8142 + 20.2612i) q^{95} +(61.7483 - 19.0424i) q^{96} +(-19.9853 - 11.5385i) q^{97} +(-56.3453 - 55.5101i) q^{98} +(-16.0742 + 9.28045i) 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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.42473 + 1.40361i 0.712367 + 0.701807i
\(3\) −1.00965 + 1.74877i −0.336551 + 0.582924i −0.983782 − 0.179371i \(-0.942594\pi\)
0.647230 + 0.762295i \(0.275927\pi\)
\(4\) 0.0597334 + 3.99955i 0.0149333 + 0.999888i
\(5\) 3.69520 + 3.36831i 0.739039 + 0.673662i
\(6\) −3.89309 + 1.07437i −0.648848 + 0.179062i
\(7\) 3.07442 0.439202 0.219601 − 0.975590i \(-0.429524\pi\)
0.219601 + 0.975590i \(0.429524\pi\)
\(8\) −5.52873 + 5.78214i −0.691091 + 0.722768i
\(9\) 2.46120 + 4.26292i 0.273466 + 0.473658i
\(10\) 0.536862 + 9.98558i 0.0536862 + 0.998558i
\(11\) 3.77070i 0.342791i 0.985202 + 0.171396i \(0.0548276\pi\)
−0.985202 + 0.171396i \(0.945172\pi\)
\(12\) −7.05462 − 3.93371i −0.587885 − 0.327809i
\(13\) 9.11726 − 5.26385i 0.701328 − 0.404912i −0.106514 − 0.994311i \(-0.533969\pi\)
0.807842 + 0.589399i \(0.200636\pi\)
\(14\) 4.38023 + 4.31530i 0.312873 + 0.308235i
\(15\) −9.62128 + 3.06123i −0.641419 + 0.204082i
\(16\) −15.9929 + 0.477814i −0.999554 + 0.0298633i
\(17\) −9.93069 − 5.73349i −0.584158 − 0.337264i 0.178626 − 0.983917i \(-0.442835\pi\)
−0.762784 + 0.646653i \(0.776168\pi\)
\(18\) −2.47694 + 9.52810i −0.137608 + 0.529339i
\(19\) 16.4485 − 9.51032i 0.865712 − 0.500543i
\(20\) −13.2510 + 14.9803i −0.662551 + 0.749017i
\(21\) −3.10410 + 5.37645i −0.147814 + 0.256022i
\(22\) −5.29261 + 5.37225i −0.240573 + 0.244193i
\(23\) −6.96607 − 12.0656i −0.302872 − 0.524591i 0.673913 − 0.738811i \(-0.264612\pi\)
−0.976785 + 0.214220i \(0.931279\pi\)
\(24\) −4.52955 − 15.5064i −0.188731 − 0.646102i
\(25\) 2.30896 + 24.8931i 0.0923583 + 0.995726i
\(26\) 20.3781 + 5.29753i 0.783773 + 0.203751i
\(27\) −28.1136 −1.04124
\(28\) 0.183645 + 12.2963i 0.00655876 + 0.439153i
\(29\) −9.09921 − 15.7603i −0.313766 − 0.543459i 0.665408 − 0.746480i \(-0.268257\pi\)
−0.979174 + 0.203021i \(0.934924\pi\)
\(30\) −18.0045 − 9.14313i −0.600151 − 0.304771i
\(31\) 50.1003i 1.61614i 0.589088 + 0.808069i \(0.299487\pi\)
−0.589088 + 0.808069i \(0.700513\pi\)
\(32\) −23.4562 − 21.7671i −0.733008 − 0.680221i
\(33\) −6.59410 − 3.80710i −0.199821 − 0.115367i
\(34\) −6.10099 − 22.1076i −0.179441 − 0.650222i
\(35\) 11.3606 + 10.3556i 0.324588 + 0.295874i
\(36\) −16.9028 + 10.0983i −0.469521 + 0.280509i
\(37\) − 17.0262i − 0.460168i −0.973171 − 0.230084i \(-0.926100\pi\)
0.973171 − 0.230084i \(-0.0739000\pi\)
\(38\) 36.7836 + 9.53770i 0.967989 + 0.250992i
\(39\) 21.2587i 0.545094i
\(40\) −39.9058 + 2.74368i −0.997645 + 0.0685920i
\(41\) 35.7872 − 61.9853i 0.872859 − 1.51184i 0.0138327 − 0.999904i \(-0.495597\pi\)
0.859026 − 0.511932i \(-0.171070\pi\)
\(42\) −11.9690 + 3.30306i −0.284976 + 0.0786443i
\(43\) −2.56904 + 4.44971i −0.0597452 + 0.103482i −0.894351 − 0.447366i \(-0.852362\pi\)
0.834606 + 0.550848i \(0.185695\pi\)
\(44\) −15.0811 + 0.225237i −0.342753 + 0.00511902i
\(45\) −5.26423 + 24.0424i −0.116983 + 0.534276i
\(46\) 7.01063 − 26.9679i 0.152405 − 0.586259i
\(47\) −27.8127 − 48.1730i −0.591760 − 1.02496i −0.993995 − 0.109422i \(-0.965100\pi\)
0.402236 − 0.915536i \(-0.368233\pi\)
\(48\) 15.3117 − 28.4503i 0.318993 − 0.592715i
\(49\) −39.5480 −0.807101
\(50\) −31.6507 + 38.7070i −0.633015 + 0.774140i
\(51\) 20.0531 − 11.5777i 0.393198 − 0.227013i
\(52\) 21.5977 + 36.1506i 0.415340 + 0.695203i
\(53\) −21.0636 + 12.1611i −0.397427 + 0.229455i −0.685373 − 0.728192i \(-0.740361\pi\)
0.287946 + 0.957647i \(0.407028\pi\)
\(54\) −40.0544 − 39.4607i −0.741748 − 0.730753i
\(55\) −12.7009 + 13.9335i −0.230925 + 0.253336i
\(56\) −16.9976 + 17.7767i −0.303529 + 0.317441i
\(57\) 0.0240704 + 38.3668i 0.000422287 + 0.673102i
\(58\) 9.15743 − 35.2260i 0.157887 − 0.607345i
\(59\) 79.5696 + 45.9396i 1.34864 + 0.778636i 0.988057 − 0.154091i \(-0.0492450\pi\)
0.360581 + 0.932728i \(0.382578\pi\)
\(60\) −12.8183 − 38.2980i −0.213638 − 0.638299i
\(61\) −2.51799 − 4.36129i −0.0412786 − 0.0714966i 0.844648 − 0.535322i \(-0.179810\pi\)
−0.885927 + 0.463826i \(0.846476\pi\)
\(62\) −70.3214 + 71.3795i −1.13422 + 1.15128i
\(63\) 7.56675 + 13.1060i 0.120107 + 0.208032i
\(64\) −2.86635 − 63.9358i −0.0447867 − 0.998997i
\(65\) 51.4204 + 11.2588i 0.791083 + 0.173212i
\(66\) −4.05113 − 14.6797i −0.0613807 − 0.222419i
\(67\) −16.9406 − 29.3419i −0.252844 − 0.437939i 0.711464 − 0.702723i \(-0.248033\pi\)
−0.964308 + 0.264784i \(0.914699\pi\)
\(68\) 22.3382 − 40.0608i 0.328503 − 0.589130i
\(69\) 28.1333 0.407728
\(70\) 1.65054 + 30.6998i 0.0235791 + 0.438569i
\(71\) 112.787 + 65.1173i 1.58854 + 0.917146i 0.993548 + 0.113417i \(0.0361795\pi\)
0.594996 + 0.803729i \(0.297154\pi\)
\(72\) −38.2561 − 9.33752i −0.531335 − 0.129688i
\(73\) 69.2413 + 39.9765i 0.948512 + 0.547623i 0.892618 − 0.450813i \(-0.148866\pi\)
0.0558933 + 0.998437i \(0.482199\pi\)
\(74\) 23.8982 − 24.2578i 0.322949 − 0.327808i
\(75\) −45.8637 − 21.0956i −0.611516 − 0.281275i
\(76\) 39.0196 + 65.2187i 0.513415 + 0.858140i
\(77\) 11.5927i 0.150555i
\(78\) −29.8390 + 30.2880i −0.382551 + 0.388307i
\(79\) 6.25869 + 3.61346i 0.0792239 + 0.0457400i 0.539089 − 0.842249i \(-0.318769\pi\)
−0.459865 + 0.887989i \(0.652102\pi\)
\(80\) −60.7062 − 52.1033i −0.758828 − 0.651292i
\(81\) 6.23423 − 10.7980i 0.0769658 − 0.133309i
\(82\) 137.991 − 38.0811i 1.68281 − 0.464403i
\(83\) 61.2186 0.737574 0.368787 − 0.929514i \(-0.379773\pi\)
0.368787 + 0.929514i \(0.379773\pi\)
\(84\) −21.6888 − 12.0939i −0.258200 − 0.143974i
\(85\) −17.3837 − 54.6360i −0.204514 − 0.642777i
\(86\) −9.90588 + 2.73371i −0.115185 + 0.0317873i
\(87\) 36.7482 0.422393
\(88\) −21.8027 − 20.8472i −0.247758 − 0.236900i
\(89\) 14.2293 + 24.6458i 0.159879 + 0.276919i 0.934825 − 0.355109i \(-0.115556\pi\)
−0.774946 + 0.632028i \(0.782223\pi\)
\(90\) −41.2464 + 26.8651i −0.458293 + 0.298501i
\(91\) 28.0303 − 16.1833i 0.308025 − 0.177838i
\(92\) 47.8408 − 28.5819i 0.520009 − 0.310673i
\(93\) −87.6139 − 50.5839i −0.942085 − 0.543913i
\(94\) 27.9906 − 107.672i 0.297773 − 1.14545i
\(95\) 92.8142 + 20.2612i 0.976992 + 0.213276i
\(96\) 61.7483 − 19.0424i 0.643211 − 0.198359i
\(97\) −19.9853 − 11.5385i −0.206034 − 0.118954i 0.393433 − 0.919353i \(-0.371287\pi\)
−0.599467 + 0.800399i \(0.704621\pi\)
\(98\) −56.3453 − 55.5101i −0.574952 − 0.566429i
\(99\) −16.0742 + 9.28045i −0.162366 + 0.0937419i
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.239.87 yes 232
4.3 odd 2 inner 380.3.p.a.239.106 yes 232
5.4 even 2 inner 380.3.p.a.239.30 yes 232
19.7 even 3 inner 380.3.p.a.159.11 ✓ 232
20.19 odd 2 inner 380.3.p.a.239.11 yes 232
76.7 odd 6 inner 380.3.p.a.159.30 yes 232
95.64 even 6 inner 380.3.p.a.159.106 yes 232
380.159 odd 6 inner 380.3.p.a.159.87 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.11 ✓ 232 19.7 even 3 inner
380.3.p.a.159.30 yes 232 76.7 odd 6 inner
380.3.p.a.159.87 yes 232 380.159 odd 6 inner
380.3.p.a.159.106 yes 232 95.64 even 6 inner
380.3.p.a.239.11 yes 232 20.19 odd 2 inner
380.3.p.a.239.30 yes 232 5.4 even 2 inner
380.3.p.a.239.87 yes 232 1.1 even 1 trivial
380.3.p.a.239.106 yes 232 4.3 odd 2 inner