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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 311.40
Character \(\chi\) \(=\) 380.311
Dual form 380.3.q.a.11.40

$q$-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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.208746 − 1.98908i −0.104373 − 0.994538i
\(3\) 2.81695 − 1.62637i 0.938985 − 0.542123i 0.0493429 − 0.998782i \(-0.484287\pi\)
0.889642 + 0.456659i \(0.150954\pi\)
\(4\) −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 \(0.413942\pi\)
−0.267078 + 0.963675i \(0.586058\pi\)
\(8\) 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 \(-0.754420\pi\)
0.716857 − 0.697220i \(-0.245580\pi\)
\(12\) −9.67175 + 8.70300i −0.805979 + 0.725250i
\(13\) −4.04092 + 6.99908i −0.310840 + 0.538391i −0.978544 − 0.206036i \(-0.933944\pi\)
0.667704 + 0.744426i \(0.267277\pi\)
\(14\) 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 0.998152 − 0.0607681i \(-0.0193550\pi\)
−0.551703 + 0.834041i \(0.686022\pi\)
\(18\) −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 0.892474 − 0.451100i \(-0.148968\pi\)
0.0555731 + 0.998455i \(0.482301\pi\)
\(24\) 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 −0.232450 − 0.972608i \(-0.574674\pi\)
0.958528 − 0.284997i \(-0.0919926\pi\)
\(30\) 5.91875 − 13.2881i 0.197292 − 0.442938i
\(31\) 1.73949i 0.0561127i 0.999606 + 0.0280564i \(0.00893179\pi\)
−0.999606 + 0.0280564i \(0.991068\pi\)
\(32\) −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 0.762152 − 0.647399i \(-0.224143\pi\)
−0.941739 + 0.336343i \(0.890810\pi\)
\(42\) 71.0141 − 51.5778i 1.69081 − 1.22804i
\(43\) −56.3816 + 32.5519i −1.31120 + 0.757021i −0.982295 − 0.187342i \(-0.940013\pi\)
−0.328904 + 0.944363i \(0.606679\pi\)
\(44\) 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 0.460358 − 0.887733i \(-0.347721\pi\)
−0.538621 + 0.842548i \(0.681054\pi\)
\(48\) 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 −0.228787 − 0.973476i \(-0.573476\pi\)
0.957449 − 0.288603i \(-0.0931907\pi\)
\(54\) 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 −0.775120 − 0.631815i \(-0.782310\pi\)
0.159608 + 0.987181i \(0.448977\pi\)
\(60\) −27.6666 − 8.99901i −0.461111 − 0.149983i
\(61\) 8.66465 − 15.0076i 0.142043 − 0.246027i −0.786223 − 0.617943i \(-0.787966\pi\)
0.928266 + 0.371917i \(0.121299\pi\)
\(62\) 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 0.963271 − 0.268531i \(-0.0865383\pi\)
0.249080 + 0.968483i \(0.419872\pi\)
\(68\) −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 −0.955393 − 0.295336i \(-0.904568\pi\)
−0.221928 + 0.975063i \(0.571235\pi\)
\(72\) 12.3650 + 2.63433i 0.171736 + 0.0365879i
\(73\) −27.9995 − 48.4966i −0.383555 − 0.664336i 0.608013 − 0.793927i \(-0.291967\pi\)
−0.991568 + 0.129591i \(0.958634\pi\)
\(74\) −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 −0.233898 − 0.972261i \(-0.575148\pi\)
0.725054 + 0.688692i \(0.241815\pi\)
\(80\) 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 \(-0.834752\pi\)
0.868245 − 0.496136i \(-0.165248\pi\)
\(84\) −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 −0.517182 − 0.855876i \(-0.673019\pi\)
0.999801 + 0.0199548i \(0.00635223\pi\)
\(90\) −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 0.893378 − 0.449305i \(-0.148328\pi\)
−0.835799 + 0.549036i \(0.814995\pi\)
\(98\) 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.311.40 yes 160
4.3 odd 2 inner 380.3.q.a.311.14 yes 160
19.11 even 3 inner 380.3.q.a.11.14 ✓ 160
76.11 odd 6 inner 380.3.q.a.11.40 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.q.a.11.14 ✓ 160 19.11 even 3 inner
380.3.q.a.11.40 yes 160 76.11 odd 6 inner
380.3.q.a.311.14 yes 160 4.3 odd 2 inner
380.3.q.a.311.40 yes 160 1.1 even 1 trivial