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Properties

Label 380.3.q.a.11.14
Level $380$
Weight $3$
Character 380.11
Analytic conductor $10.354$
Analytic rank $0$
Dimension $160$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

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 11.14
Character \(\chi\) \(=\) 380.11
Dual form 380.3.q.a.311.14

$q$-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+(-1.61822 + 1.17532i) q^{2} +(-2.81695 - 1.62637i) q^{3} +(1.23726 - 3.80384i) q^{4} +(1.11803 - 1.93649i) q^{5} +(6.46995 - 0.678995i) q^{6} +13.4914i q^{7} +(2.46856 + 7.60961i) q^{8} +(0.790156 + 1.36859i) q^{9} +(0.466769 + 4.44771i) q^{10} -15.3388i q^{11} +(-9.67175 + 8.70300i) q^{12} +(-4.04092 - 6.99908i) q^{13} +(-15.8567 - 21.8321i) q^{14} +(-6.29890 + 3.63667i) q^{15} +(-12.9384 - 9.41266i) 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} +(18.0280 + 24.8216i) q^{22} +(-21.8051 + 12.5892i) q^{23} +(5.42221 - 25.4507i) q^{24} +(-2.50000 - 4.33013i) q^{25} +(14.7652 + 6.57667i) q^{26} +24.1343i q^{27} +(51.3193 + 16.6924i) q^{28} +(21.0563 + 36.4706i) q^{29} +(5.91875 - 13.2881i) q^{30} +1.73949i q^{31} +(32.0000 + 0.0250387i) q^{32} +(-24.9466 + 43.2088i) q^{33} +(3.16861 + 30.1927i) q^{34} +(26.1261 + 15.0839i) q^{35} +(6.18352 - 1.31233i) q^{36} +23.8583 q^{37} +(-8.68786 - 36.9935i) q^{38} +26.2881i q^{39} +(17.4959 + 3.72745i) q^{40} +(-7.36310 + 12.7533i) q^{41} +(9.16063 + 87.2890i) q^{42} +(56.3816 + 32.5519i) q^{43} +(-58.3465 - 18.9781i) q^{44} +3.53368 q^{45} +(20.4891 - 45.9999i) q^{46} +(3.67835 - 2.12370i) q^{47} +(21.1384 + 47.5576i) q^{48} -133.019 q^{49} +(9.13482 + 4.06880i) q^{50} +(-42.7593 + 24.6871i) q^{51} +(-31.6230 + 6.71134i) q^{52} +(38.6191 + 66.8902i) q^{53} +(-28.3655 - 39.0546i) 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} +(6.03995 + 28.4595i) q^{60} +(8.66465 + 15.0076i) q^{61} +(-2.04446 - 2.81488i) q^{62} +(-18.4643 + 10.6603i) q^{63} +(-51.8124 + 37.5696i) q^{64} -18.0715 q^{65} +(-10.4150 - 99.2415i) q^{66} +(-81.2275 + 46.8967i) q^{67} +(-40.6135 - 45.1343i) q^{68} +81.8985 q^{69} +(-60.0061 + 6.29740i) q^{70} +(83.5898 + 48.2606i) q^{71} +(-8.46389 + 9.39123i) q^{72} +(-27.9995 + 48.4966i) q^{73} +(-38.6080 + 28.0411i) q^{74} +16.2637i q^{75} +(57.5380 + 49.6526i) q^{76} +206.943 q^{77} +(-30.8969 - 42.5399i) q^{78} +(-38.8013 - 22.4019i) q^{79} +(-32.6931 + 14.5314i) q^{80} +(46.3627 - 80.3026i) q^{81} +(-3.07403 - 29.2915i) q^{82} -82.3586i q^{83} +(-117.416 - 130.486i) q^{84} +(-16.9709 - 29.3945i) q^{85} +(-129.496 + 13.5901i) q^{86} -136.981i q^{87} +(116.723 - 37.8649i) q^{88} +(42.9531 + 74.3969i) q^{89} +(-5.71827 + 4.15320i) q^{90} +(94.4277 - 54.5179i) q^{91} +(20.9086 + 98.5190i) 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} +(215.254 - 156.340i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.61822 + 1.17532i −0.809109 + 0.587659i
\(3\) −2.81695 − 1.62637i −0.938985 − 0.542123i −0.0493429 − 0.998782i \(-0.515713\pi\)
−0.889642 + 0.456659i \(0.849046\pi\)
\(4\) 1.23726 − 3.80384i 0.309315 − 0.950960i
\(5\) 1.11803 − 1.93649i 0.223607 − 0.387298i
\(6\) 6.46995 − 0.678995i 1.07832 − 0.113166i
\(7\) 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
\(10\) 0.466769 + 4.44771i 0.0466769 + 0.444771i
\(11\) − 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 0.667704 − 0.744426i \(-0.267277\pi\)
−0.978544 + 0.206036i \(0.933944\pi\)
\(14\) −15.8567 − 21.8321i −1.13262 − 1.55944i
\(15\) −6.29890 + 3.63667i −0.419927 + 0.242445i
\(16\) −12.9384 − 9.41266i −0.808649 − 0.588292i
\(17\) 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
\(22\) 18.0280 + 24.8216i 0.819455 + 1.12825i
\(23\) −21.8051 + 12.5892i −0.948047 + 0.547355i −0.892474 − 0.451100i \(-0.851032\pi\)
−0.0555731 + 0.998455i \(0.517699\pi\)
\(24\) 5.42221 − 25.4507i 0.225925 − 1.06045i
\(25\) −2.50000 − 4.33013i −0.100000 − 0.173205i
\(26\) 14.7652 + 6.57667i 0.567893 + 0.252949i
\(27\) 24.1343i 0.893863i
\(28\) 51.3193 + 16.6924i 1.83283 + 0.596157i
\(29\) 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 \(0.00893179\pi\)
−0.999606 + 0.0280564i \(0.991068\pi\)
\(32\) 32.0000 + 0.0250387i 1.00000 + 0.000782460i
\(33\) −24.9466 + 43.2088i −0.755958 + 1.30936i
\(34\) 3.16861 + 30.1927i 0.0931943 + 0.888022i
\(35\) 26.1261 + 15.0839i 0.746459 + 0.430968i
\(36\) 6.18352 − 1.31233i 0.171765 − 0.0364535i
\(37\) 23.8583 0.644820 0.322410 − 0.946600i \(-0.395507\pi\)
0.322410 + 0.946600i \(0.395507\pi\)
\(38\) −8.68786 − 36.9935i −0.228628 − 0.973514i
\(39\) 26.2881i 0.674054i
\(40\) 17.4959 + 3.72745i 0.437397 + 0.0931863i
\(41\) −7.36310 + 12.7533i −0.179588 + 0.311055i −0.941739 − 0.336343i \(-0.890810\pi\)
0.762152 + 0.647399i \(0.224143\pi\)
\(42\) 9.16063 + 87.2890i 0.218110 + 2.07831i
\(43\) 56.3816 + 32.5519i 1.31120 + 0.757021i 0.982295 − 0.187342i \(-0.0599873\pi\)
0.328904 + 0.944363i \(0.393321\pi\)
\(44\) −58.3465 − 18.9781i −1.32606 − 0.431321i
\(45\) 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 \(-0.652279\pi\)
0.538621 + 0.842548i \(0.318946\pi\)
\(48\) 21.1384 + 47.5576i 0.440383 + 0.990784i
\(49\) −133.019 −2.71468
\(50\) 9.13482 + 4.06880i 0.182696 + 0.0813759i
\(51\) −42.7593 + 24.6871i −0.838418 + 0.484061i
\(52\) −31.6230 + 6.71134i −0.608135 + 0.129064i
\(53\) 38.6191 + 66.8902i 0.728662 + 1.26208i 0.957449 + 0.288603i \(0.0931907\pi\)
−0.228787 + 0.973476i \(0.573476\pi\)
\(54\) −28.3655 − 39.0546i −0.525287 − 0.723233i
\(55\) −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 \(-0.217690\pi\)
−0.159608 + 0.987181i \(0.551023\pi\)
\(60\) 6.03995 + 28.4595i 0.100666 + 0.474325i
\(61\) 8.66465 + 15.0076i 0.142043 + 0.246027i 0.928266 − 0.371917i \(-0.121299\pi\)
−0.786223 + 0.617943i \(0.787966\pi\)
\(62\) −2.04446 − 2.81488i −0.0329751 − 0.0454013i
\(63\) −18.4643 + 10.6603i −0.293084 + 0.169212i
\(64\) −51.8124 + 37.5696i −0.809569 + 0.587025i
\(65\) −18.0715 −0.278024
\(66\) −10.4150 − 99.2415i −0.157803 − 1.50366i
\(67\) −81.2275 + 46.8967i −1.21235 + 0.699951i −0.963271 − 0.268531i \(-0.913462\pi\)
−0.249080 + 0.968483i \(0.580128\pi\)
\(68\) −40.6135 − 45.1343i −0.597258 − 0.663740i
\(69\) 81.8985 1.18694
\(70\) −60.0061 + 6.29740i −0.857229 + 0.0899628i
\(71\) 83.5898 + 48.2606i 1.17732 + 0.679727i 0.955393 − 0.295336i \(-0.0954317\pi\)
0.221928 + 0.975063i \(0.428765\pi\)
\(72\) −8.46389 + 9.39123i −0.117554 + 0.130434i
\(73\) −27.9995 + 48.4966i −0.383555 + 0.664336i −0.991568 − 0.129591i \(-0.958634\pi\)
0.608013 + 0.793927i \(0.291967\pi\)
\(74\) −38.6080 + 28.0411i −0.521729 + 0.378934i
\(75\) 16.2637i 0.216849i
\(76\) 57.5380 + 49.6526i 0.757079 + 0.653324i
\(77\) 206.943 2.68757
\(78\) −30.8969 − 42.5399i −0.396114 − 0.545383i
\(79\) −38.8013 − 22.4019i −0.491156 − 0.283569i 0.233898 − 0.972261i \(-0.424852\pi\)
−0.725054 + 0.688692i \(0.758185\pi\)
\(80\) −32.6931 + 14.5314i −0.408664 + 0.181642i
\(81\) 46.3627 − 80.3026i 0.572379 − 0.991390i
\(82\) −3.07403 − 29.2915i −0.0374882 − 0.357214i
\(83\) − 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
\(86\) −129.496 + 13.5901i −1.50577 + 0.158025i
\(87\) − 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 \(0.673019\pi\)
\(90\) −5.71827 + 4.15320i −0.0635363 + 0.0461467i
\(91\) 94.4277 − 54.5179i 1.03767 − 0.599097i
\(92\) 20.9086 + 98.5190i 0.227268 + 1.07086i
\(93\) 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 \(0.148328\pi\)
\(98\) 215.254 − 156.340i 2.19647 − 1.59530i
\(99\) 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.11.14 ✓ 160
4.3 odd 2 inner 380.3.q.a.11.40 yes 160
19.7 even 3 inner 380.3.q.a.311.40 yes 160
76.7 odd 6 inner 380.3.q.a.311.14 yes 160
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.q.a.11.14 ✓ 160 1.1 even 1 trivial
380.3.q.a.11.40 yes 160 4.3 odd 2 inner
380.3.q.a.311.14 yes 160 76.7 odd 6 inner
380.3.q.a.311.40 yes 160 19.7 even 3 inner