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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 159.30
Character \(\chi\) \(=\) 380.159
Dual form 380.3.p.a.239.30

$q$-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} +(-4.76464 + 1.51598i) q^{5} +(-3.89309 - 1.07437i) q^{6} -3.07442 q^{7} +(5.52873 + 5.78214i) q^{8} +(2.46120 - 4.26292i) q^{9} +(4.66050 - 8.84759i) q^{10} -3.77070i q^{11} +(7.05462 - 3.93371i) q^{12} +(-9.11726 - 5.26385i) q^{13} +(4.38023 - 4.31530i) q^{14} +(-7.46174 - 6.80166i) 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} +(5.77863 + 19.1470i) 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} +(20.4036 - 14.4462i) q^{25} +(20.3781 - 5.29753i) q^{26} +28.1136 q^{27} +(-0.183645 + 12.2963i) q^{28} +(-9.09921 + 15.7603i) q^{29} +(20.1779 - 0.782853i) q^{30} -50.1003i q^{31} +(23.4562 - 21.7671i) q^{32} +(6.59410 - 3.80710i) q^{33} +(-6.10099 + 22.1076i) q^{34} +(14.6485 - 4.66075i) q^{35} +(-16.9028 - 10.0983i) q^{36} -17.0262i q^{37} +(-36.7836 + 9.53770i) q^{38} -21.2587i q^{39} +(-35.1080 - 19.1684i) 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} +(-8.79285 + 49.2208i) 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} +(5.71630 + 17.9660i) 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} +(-27.6493 + 29.4373i) 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} +(-14.3283 + 27.2012i) 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} +(76.9246 - 21.9682i) q^{80} +(6.23423 + 10.7980i) q^{81} +(-137.991 - 38.0811i) q^{82} -61.2186 q^{83} +(-21.6888 + 12.0939i) q^{84} +(-38.6243 + 42.3727i) q^{85} +(-9.90588 - 2.73371i) q^{86} -36.7482 q^{87} +(21.8027 - 20.8472i) q^{88} +(14.2293 - 24.6458i) q^{89} +(-26.2461 - 41.6430i) 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} +(-92.7887 - 20.3777i) 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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.42473 + 1.40361i −0.712367 + 0.701807i
\(3\) 1.00965 + 1.74877i 0.336551 + 0.582924i 0.983782 − 0.179371i \(-0.0574061\pi\)
−0.647230 + 0.762295i \(0.724073\pi\)
\(4\) 0.0597334 − 3.99955i 0.0149333 − 0.999888i
\(5\) −4.76464 + 1.51598i −0.952928 + 0.303196i
\(6\) −3.89309 − 1.07437i −0.648848 − 0.179062i
\(7\) −3.07442 −0.439202 −0.219601 − 0.975590i \(-0.570476\pi\)
−0.219601 + 0.975590i \(0.570476\pi\)
\(8\) 5.52873 + 5.78214i 0.691091 + 0.722768i
\(9\) 2.46120 − 4.26292i 0.273466 − 0.473658i
\(10\) 4.66050 − 8.84759i 0.466050 − 0.884759i
\(11\) − 3.77070i − 0.342791i −0.985202 − 0.171396i \(-0.945172\pi\)
0.985202 − 0.171396i \(-0.0548276\pi\)
\(12\) 7.05462 − 3.93371i 0.587885 − 0.327809i
\(13\) −9.11726 − 5.26385i −0.701328 − 0.404912i 0.106514 − 0.994311i \(-0.466031\pi\)
−0.807842 + 0.589399i \(0.799364\pi\)
\(14\) 4.38023 − 4.31530i 0.312873 − 0.308235i
\(15\) −7.46174 − 6.80166i −0.497449 − 0.453444i
\(16\) −15.9929 − 0.477814i −0.999554 − 0.0298633i
\(17\) 9.93069 − 5.73349i 0.584158 − 0.337264i −0.178626 − 0.983917i \(-0.557165\pi\)
0.762784 + 0.646653i \(0.223832\pi\)
\(18\) 2.47694 + 9.52810i 0.137608 + 0.529339i
\(19\) 16.4485 + 9.51032i 0.865712 + 0.500543i
\(20\) 5.77863 + 19.1470i 0.288931 + 0.957350i
\(21\) −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 \(-0.735388\pi\)
0.976785 + 0.214220i \(0.0687210\pi\)
\(24\) −4.52955 + 15.5064i −0.188731 + 0.646102i
\(25\) 20.4036 − 14.4462i 0.816145 − 0.577848i
\(26\) 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 −0.979174 − 0.203021i \(-0.934924\pi\)
0.665408 + 0.746480i \(0.268257\pi\)
\(30\) 20.1779 − 0.782853i 0.672597 − 0.0260951i
\(31\) − 50.1003i − 1.61614i −0.589088 − 0.808069i \(-0.700513\pi\)
0.589088 − 0.808069i \(-0.299487\pi\)
\(32\) 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
\(35\) 14.6485 − 4.66075i 0.418528 − 0.133164i
\(36\) −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
\(40\) −35.1080 − 19.1684i −0.877700 − 0.479210i
\(41\) 35.7872 + 61.9853i 0.872859 + 1.51184i 0.859026 + 0.511932i \(0.171070\pi\)
0.0138327 + 0.999904i \(0.495597\pi\)
\(42\) 11.9690 + 3.30306i 0.284976 + 0.0786443i
\(43\) 2.56904 + 4.44971i 0.0597452 + 0.103482i 0.894351 − 0.447366i \(-0.147638\pi\)
−0.834606 + 0.550848i \(0.814305\pi\)
\(44\) −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 −0.402236 − 0.915536i \(-0.631767\pi\)
0.993995 − 0.109422i \(-0.0348999\pi\)
\(48\) −15.3117 − 28.4503i −0.318993 − 0.592715i
\(49\) −39.5480 −0.807101
\(50\) −8.79285 + 49.2208i −0.175857 + 0.984416i
\(51\) 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 \(-0.259639\pi\)
−0.287946 + 0.957647i \(0.592972\pi\)
\(54\) −40.0544 + 39.4607i −0.741748 + 0.730753i
\(55\) 5.71630 + 17.9660i 0.103933 + 0.326655i
\(56\) −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 0.360581 − 0.932728i \(-0.382578\pi\)
0.988057 + 0.154091i \(0.0492450\pi\)
\(60\) −27.6493 + 29.4373i −0.460822 + 0.490622i
\(61\) −2.51799 + 4.36129i −0.0412786 + 0.0714966i −0.885927 − 0.463826i \(-0.846476\pi\)
0.844648 + 0.535322i \(0.179810\pi\)
\(62\) 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 \(-0.751967\pi\)
0.964308 + 0.264784i \(0.0853007\pi\)
\(68\) −22.3382 − 40.0608i −0.328503 − 0.589130i
\(69\) 28.1333 0.407728
\(70\) −14.3283 + 27.2012i −0.204690 + 0.388588i
\(71\) 112.787 − 65.1173i 1.58854 − 0.917146i 0.594996 − 0.803729i \(-0.297154\pi\)
0.993548 − 0.113417i \(-0.0361795\pi\)
\(72\) 38.2561 − 9.33752i 0.531335 − 0.129688i
\(73\) −69.2413 + 39.9765i −0.948512 + 0.547623i −0.892618 − 0.450813i \(-0.851134\pi\)
−0.0558933 + 0.998437i \(0.517801\pi\)
\(74\) 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 −0.459865 − 0.887989i \(-0.652102\pi\)
0.539089 + 0.842249i \(0.318769\pi\)
\(80\) 76.9246 − 21.9682i 0.961558 − 0.274603i
\(81\) 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 \(-0.620227\pi\)
−0.368787 + 0.929514i \(0.620227\pi\)
\(84\) −21.6888 + 12.0939i −0.258200 + 0.143974i
\(85\) −38.6243 + 42.3727i −0.454404 + 0.498503i
\(86\) −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 −0.774946 − 0.632028i \(-0.782223\pi\)
0.934825 + 0.355109i \(0.115556\pi\)
\(90\) −26.2461 − 41.6430i −0.291624 − 0.462700i
\(91\) 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
\(95\) −92.7887 − 20.3777i −0.976724 − 0.214502i
\(96\) 61.7483 + 19.0424i 0.643211 + 0.198359i
\(97\) 19.9853 − 11.5385i 0.206034 − 0.118954i −0.393433 − 0.919353i \(-0.628713\pi\)
0.599467 + 0.800399i \(0.295379\pi\)
\(98\) 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.159.30 yes 232
4.3 odd 2 inner 380.3.p.a.159.11 ✓ 232
5.4 even 2 inner 380.3.p.a.159.87 yes 232
19.11 even 3 inner 380.3.p.a.239.106 yes 232
20.19 odd 2 inner 380.3.p.a.159.106 yes 232
76.11 odd 6 inner 380.3.p.a.239.87 yes 232
95.49 even 6 inner 380.3.p.a.239.11 yes 232
380.239 odd 6 inner 380.3.p.a.239.30 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.p.a.159.11 ✓ 232 4.3 odd 2 inner
380.3.p.a.159.30 yes 232 1.1 even 1 trivial
380.3.p.a.159.87 yes 232 5.4 even 2 inner
380.3.p.a.159.106 yes 232 20.19 odd 2 inner
380.3.p.a.239.11 yes 232 95.49 even 6 inner
380.3.p.a.239.30 yes 232 380.239 odd 6 inner
380.3.p.a.239.87 yes 232 76.11 odd 6 inner
380.3.p.a.239.106 yes 232 19.11 even 3 inner