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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(11,380)] 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("380.11"); 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, 0, 4])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.q (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: \(160\)
Relative dimension: \(80\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 11.40
Character \(\chi\) \(=\) 380.11
Dual form 380.3.q.a.311.40

$q$-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+(-0.208746 + 1.98908i) q^{2} +(2.81695 + 1.62637i) q^{3} +(-3.91285 - 0.830422i) q^{4} +(1.11803 - 1.93649i) q^{5} +(-3.82300 + 5.26364i) q^{6} -13.4914i q^{7} +(2.46856 - 7.60961i) q^{8} +(0.790156 + 1.36859i) q^{9} +(3.61845 + 2.62809i) q^{10} +15.3388i q^{11} +(-9.67175 - 8.70300i) q^{12} +(-4.04092 - 6.99908i) q^{13} +(26.8355 + 2.81628i) q^{14} +(6.29890 - 3.63667i) q^{15} +(14.6208 + 6.49864i) q^{16} +(7.58964 - 13.1456i) q^{17} +(-2.88717 + 1.28599i) q^{18} +(7.35507 - 17.5186i) q^{19} +(-5.98281 + 6.64876i) q^{20} +(21.9421 - 38.0048i) q^{21} +(-30.5101 - 3.20192i) q^{22} +(21.8051 - 12.5892i) q^{23} +(19.3299 - 17.4211i) q^{24} +(-2.50000 - 4.33013i) q^{25} +(14.7652 - 6.57667i) q^{26} -24.1343i q^{27} +(-11.2036 + 52.7900i) q^{28} +(21.0563 + 36.4706i) q^{29} +(5.91875 + 13.2881i) q^{30} -1.73949i q^{31} +(-15.9783 + 27.7253i) q^{32} +(-24.9466 + 43.2088i) q^{33} +(24.5634 + 17.8405i) q^{34} +(-26.1261 - 15.0839i) q^{35} +(-1.95525 - 6.01125i) q^{36} +23.8583 q^{37} +(33.3106 + 18.2867i) q^{38} -26.2881i q^{39} +(-11.9760 - 13.2882i) q^{40} +(-7.36310 + 12.7533i) q^{41} +(71.0141 + 51.5778i) q^{42} +(-56.3816 - 32.5519i) q^{43} +(12.7377 - 60.0186i) q^{44} +3.53368 q^{45} +(20.4891 + 45.9999i) q^{46} +(-3.67835 + 2.12370i) q^{47} +(30.6169 + 42.0852i) q^{48} -133.019 q^{49} +(9.13482 - 4.06880i) q^{50} +(42.7593 - 24.6871i) q^{51} +(9.99933 + 30.7420i) q^{52} +(38.6191 + 66.8902i) q^{53} +(48.0050 + 5.03793i) q^{54} +(29.7035 + 17.1493i) q^{55} +(-102.665 - 33.3045i) q^{56} +(49.2107 - 37.3872i) q^{57} +(-76.9381 + 34.2695i) q^{58} +(-36.3152 - 20.9666i) q^{59} +(-27.6666 + 8.99901i) q^{60} +(8.66465 + 15.0076i) q^{61} +(3.45999 + 0.363112i) q^{62} +(18.4643 - 10.6603i) q^{63} +(-51.8124 - 37.5696i) q^{64} -18.0715 q^{65} +(-80.7381 - 58.6404i) q^{66} +(81.2275 - 46.8967i) q^{67} +(-40.6135 + 45.1343i) q^{68} +81.8985 q^{69} +(35.4567 - 48.8181i) q^{70} +(-83.5898 - 48.2606i) q^{71} +(12.3650 - 2.63433i) q^{72} +(-27.9995 + 48.4966i) q^{73} +(-4.98032 + 47.4560i) q^{74} -16.2637i q^{75} +(-43.3272 + 62.4400i) q^{76} +206.943 q^{77} +(52.2891 + 5.48753i) q^{78} +(38.8013 + 22.4019i) q^{79} +(28.9311 - 21.0474i) q^{80} +(46.3627 - 80.3026i) q^{81} +(-23.8302 - 17.3080i) q^{82} +82.3586i q^{83} +(-117.416 + 130.486i) q^{84} +(-16.9709 - 29.3945i) q^{85} +(76.5177 - 105.352i) q^{86} +136.981i q^{87} +(116.723 + 37.8649i) q^{88} +(42.9531 + 74.3969i) q^{89} +(-0.737641 + 7.02877i) q^{90} +(-94.4277 + 54.5179i) q^{91} +(-95.7743 + 31.1521i) q^{92} +(2.82906 - 4.90008i) q^{93} +(-3.45636 - 7.75983i) q^{94} +(-25.7015 - 33.8295i) q^{95} +(-90.1018 + 52.1143i) q^{96} +(5.58522 - 9.67388i) q^{97} +(27.7672 - 264.585i) q^{98} +(-20.9926 + 12.1201i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 160 q + 2 q^{4} + 6 q^{6} + 248 q^{9} - 10 q^{10} - 16 q^{13} - 14 q^{16} + 48 q^{17} + 48 q^{21} - 44 q^{24} - 400 q^{25} + 68 q^{26} + 60 q^{28} - 80 q^{30} + 30 q^{32} - 40 q^{33} - 22 q^{34} + 52 q^{36}+ \cdots - 226 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.208746 + 1.98908i −0.104373 + 0.994538i
\(3\) 2.81695 + 1.62637i 0.938985 + 0.542123i 0.889642 − 0.456659i \(-0.150954\pi\)
0.0493429 + 0.998782i \(0.484287\pi\)
\(4\) −3.91285 − 0.830422i −0.978213 − 0.207606i
\(5\) 1.11803 − 1.93649i 0.223607 − 0.387298i
\(6\) −3.82300 + 5.26364i −0.637167 + 0.877273i
\(7\) − 13.4914i − 1.92735i −0.267078 − 0.963675i \(-0.586058\pi\)
0.267078 − 0.963675i \(-0.413942\pi\)
\(8\) 2.46856 − 7.60961i 0.308570 − 0.951201i
\(9\) 0.790156 + 1.36859i 0.0877951 + 0.152066i
\(10\) 3.61845 + 2.62809i 0.361845 + 0.262809i
\(11\) 15.3388i 1.39444i 0.716857 + 0.697220i \(0.245580\pi\)
−0.716857 + 0.697220i \(0.754420\pi\)
\(12\) −9.67175 − 8.70300i −0.805979 − 0.725250i
\(13\) −4.04092 − 6.99908i −0.310840 − 0.538391i 0.667704 − 0.744426i \(-0.267277\pi\)
−0.978544 + 0.206036i \(0.933944\pi\)
\(14\) 26.8355 + 2.81628i 1.91682 + 0.201163i
\(15\) 6.29890 − 3.63667i 0.419927 − 0.242445i
\(16\) 14.6208 + 6.49864i 0.913800 + 0.406165i
\(17\) 7.58964 − 13.1456i 0.446449 − 0.773273i −0.551703 − 0.834041i \(-0.686022\pi\)
0.998152 + 0.0607681i \(0.0193550\pi\)
\(18\) −2.88717 + 1.28599i −0.160398 + 0.0714441i
\(19\) 7.35507 − 17.5186i 0.387109 − 0.922034i
\(20\) −5.98281 + 6.64876i −0.299140 + 0.332438i
\(21\) 21.9421 − 38.0048i 1.04486 − 1.80975i
\(22\) −30.5101 − 3.20192i −1.38682 − 0.145542i
\(23\) 21.8051 − 12.5892i 0.948047 − 0.547355i 0.0555731 − 0.998455i \(-0.482301\pi\)
0.892474 + 0.451100i \(0.148968\pi\)
\(24\) 19.3299 − 17.4211i 0.805411 − 0.725881i
\(25\) −2.50000 − 4.33013i −0.100000 − 0.173205i
\(26\) 14.7652 − 6.57667i 0.567893 − 0.252949i
\(27\) − 24.1343i − 0.893863i
\(28\) −11.2036 + 52.7900i −0.400128 + 1.88536i
\(29\) 21.0563 + 36.4706i 0.726079 + 1.25761i 0.958528 + 0.284997i \(0.0919926\pi\)
−0.232450 + 0.972608i \(0.574674\pi\)
\(30\) 5.91875 + 13.2881i 0.197292 + 0.442938i
\(31\) − 1.73949i − 0.0561127i −0.999606 − 0.0280564i \(-0.991068\pi\)
0.999606 − 0.0280564i \(-0.00893179\pi\)
\(32\) −15.9783 + 27.7253i −0.499322 + 0.866416i
\(33\) −24.9466 + 43.2088i −0.755958 + 1.30936i
\(34\) 24.5634 + 17.8405i 0.722452 + 0.524719i
\(35\) −26.1261 − 15.0839i −0.746459 − 0.430968i
\(36\) −1.95525 − 6.01125i −0.0543126 − 0.166979i
\(37\) 23.8583 0.644820 0.322410 − 0.946600i \(-0.395507\pi\)
0.322410 + 0.946600i \(0.395507\pi\)
\(38\) 33.3106 + 18.2867i 0.876594 + 0.481230i
\(39\) − 26.2881i − 0.674054i
\(40\) −11.9760 − 13.2882i −0.299400 − 0.332204i
\(41\) −7.36310 + 12.7533i −0.179588 + 0.311055i −0.941739 − 0.336343i \(-0.890810\pi\)
0.762152 + 0.647399i \(0.224143\pi\)
\(42\) 71.0141 + 51.5778i 1.69081 + 1.22804i
\(43\) −56.3816 − 32.5519i −1.31120 − 0.757021i −0.328904 − 0.944363i \(-0.606679\pi\)
−0.982295 + 0.187342i \(0.940013\pi\)
\(44\) 12.7377 − 60.0186i 0.289493 − 1.36406i
\(45\) 3.53368 0.0785263
\(46\) 20.4891 + 45.9999i 0.445415 + 0.999998i
\(47\) −3.67835 + 2.12370i −0.0782628 + 0.0451850i −0.538621 − 0.842548i \(-0.681054\pi\)
0.460358 + 0.887733i \(0.347721\pi\)
\(48\) 30.6169 + 42.0852i 0.637853 + 0.876775i
\(49\) −133.019 −2.71468
\(50\) 9.13482 − 4.06880i 0.182696 − 0.0813759i
\(51\) 42.7593 − 24.6871i 0.838418 − 0.484061i
\(52\) 9.99933 + 30.7420i 0.192295 + 0.591193i
\(53\) 38.6191 + 66.8902i 0.728662 + 1.26208i 0.957449 + 0.288603i \(0.0931907\pi\)
−0.228787 + 0.973476i \(0.573476\pi\)
\(54\) 48.0050 + 5.03793i 0.888981 + 0.0932950i
\(55\) 29.7035 + 17.1493i 0.540064 + 0.311806i
\(56\) −102.665 − 33.3045i −1.83330 − 0.594723i
\(57\) 49.2107 − 37.3872i 0.863345 − 0.655915i
\(58\) −76.9381 + 34.2695i −1.32652 + 0.590853i
\(59\) −36.3152 − 20.9666i −0.615512 − 0.355366i 0.159608 − 0.987181i \(-0.448977\pi\)
−0.775120 + 0.631815i \(0.782310\pi\)
\(60\) −27.6666 + 8.99901i −0.461111 + 0.149983i
\(61\) 8.66465 + 15.0076i 0.142043 + 0.246027i 0.928266 − 0.371917i \(-0.121299\pi\)
−0.786223 + 0.617943i \(0.787966\pi\)
\(62\) 3.45999 + 0.363112i 0.0558062 + 0.00585664i
\(63\) 18.4643 − 10.6603i 0.293084 − 0.169212i
\(64\) −51.8124 − 37.5696i −0.809569 − 0.587025i
\(65\) −18.0715 −0.278024
\(66\) −80.7381 − 58.6404i −1.22331 − 0.888491i
\(67\) 81.2275 − 46.8967i 1.21235 − 0.699951i 0.249080 − 0.968483i \(-0.419872\pi\)
0.963271 + 0.268531i \(0.0865383\pi\)
\(68\) −40.6135 + 45.1343i −0.597258 + 0.663740i
\(69\) 81.8985 1.18694
\(70\) 35.4567 − 48.8181i 0.506525 − 0.697401i
\(71\) −83.5898 − 48.2606i −1.17732 − 0.679727i −0.221928 − 0.975063i \(-0.571235\pi\)
−0.955393 + 0.295336i \(0.904568\pi\)
\(72\) 12.3650 − 2.63433i 0.171736 − 0.0365879i
\(73\) −27.9995 + 48.4966i −0.383555 + 0.664336i −0.991568 − 0.129591i \(-0.958634\pi\)
0.608013 + 0.793927i \(0.291967\pi\)
\(74\) −4.98032 + 47.4560i −0.0673017 + 0.641298i
\(75\) − 16.2637i − 0.216849i
\(76\) −43.3272 + 62.4400i −0.570094 + 0.821579i
\(77\) 206.943 2.68757
\(78\) 52.2891 + 5.48753i 0.670373 + 0.0703530i
\(79\) 38.8013 + 22.4019i 0.491156 + 0.283569i 0.725054 − 0.688692i \(-0.241815\pi\)
−0.233898 + 0.972261i \(0.575148\pi\)
\(80\) 28.9311 − 21.0474i 0.361639 − 0.263092i
\(81\) 46.3627 − 80.3026i 0.572379 − 0.991390i
\(82\) −23.8302 − 17.3080i −0.290612 − 0.211073i
\(83\) 82.3586i 0.992273i 0.868245 + 0.496136i \(0.165248\pi\)
−0.868245 + 0.496136i \(0.834752\pi\)
\(84\) −117.416 + 130.486i −1.39781 + 1.55340i
\(85\) −16.9709 − 29.3945i −0.199658 − 0.345818i
\(86\) 76.5177 − 105.352i 0.889740 − 1.22503i
\(87\) 136.981i 1.57450i
\(88\) 116.723 + 37.8649i 1.32639 + 0.430283i
\(89\) 42.9531 + 74.3969i 0.482619 + 0.835921i 0.999801 − 0.0199548i \(-0.00635223\pi\)
−0.517182 + 0.855876i \(0.673019\pi\)
\(90\) −0.737641 + 7.02877i −0.00819601 + 0.0780974i
\(91\) −94.4277 + 54.5179i −1.03767 + 0.599097i
\(92\) −95.7743 + 31.1521i −1.04103 + 0.338610i
\(93\) 2.82906 − 4.90008i 0.0304200 − 0.0526890i
\(94\) −3.45636 − 7.75983i −0.0367697 − 0.0825514i
\(95\) −25.7015 − 33.8295i −0.270542 − 0.356100i
\(96\) −90.1018 + 52.1143i −0.938560 + 0.542858i
\(97\) 5.58522 − 9.67388i 0.0575796 − 0.0997307i −0.835799 − 0.549036i \(-0.814995\pi\)
0.893378 + 0.449305i \(0.148328\pi\)
\(98\) 27.7672 − 264.585i 0.283338 − 2.69985i
\(99\) −20.9926 + 12.1201i −0.212046 + 0.122425i
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.q.a.11.40 yes 160
4.3 odd 2 inner 380.3.q.a.11.14 ✓ 160
19.7 even 3 inner 380.3.q.a.311.14 yes 160
76.7 odd 6 inner 380.3.q.a.311.40 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.q.a.11.14 ✓ 160 4.3 odd 2 inner
380.3.q.a.11.40 yes 160 1.1 even 1 trivial
380.3.q.a.311.14 yes 160 19.7 even 3 inner
380.3.q.a.311.40 yes 160 76.7 odd 6 inner