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Properties

Label 380.2.s.a.179.19
Level $380$
Weight $2$
Character 380.179
Analytic conductor $3.034$
Analytic rank $0$
Dimension $112$
Inner twists $8$

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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] = [380,2,Mod(179,380)] 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("380.179"); S:= CuspForms(chi, 2); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.s (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: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(56\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 179.19
Character \(\chi\) \(=\) 380.179
Dual form 380.2.s.a.259.19

$q$-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+(-0.770699 - 1.18576i) q^{2} +(2.83828 + 1.63868i) q^{3} +(-0.812047 + 1.82773i) q^{4} +(2.18912 - 0.455815i) q^{5} +(-0.244379 - 4.62845i) q^{6} -0.387143 q^{7} +(2.79308 - 0.445735i) q^{8} +(3.87057 + 6.70403i) q^{9} +(-2.22764 - 2.24447i) q^{10} +2.91567i q^{11} +(-5.29988 + 3.85692i) q^{12} +(-2.14283 - 3.71149i) q^{13} +(0.298371 + 0.459059i) q^{14} +(6.96027 + 2.29354i) q^{15} +(-2.68116 - 2.96840i) q^{16} +(-4.14205 - 2.39142i) q^{17} +(4.96631 - 9.75635i) q^{18} +(-4.27334 + 0.859404i) q^{19} +(-0.944559 + 4.37125i) q^{20} +(-1.09882 - 0.634406i) q^{21} +(3.45728 - 2.24710i) q^{22} +(-1.45437 - 2.51904i) q^{23} +(8.65799 + 3.31186i) q^{24} +(4.58446 - 1.99567i) q^{25} +(-2.74946 + 5.40132i) q^{26} +15.5385i q^{27} +(0.314378 - 0.707592i) q^{28} +(0.172278 - 0.0994646i) q^{29} +(-2.64469 - 10.0208i) q^{30} +5.67744 q^{31} +(-1.45343 + 5.46695i) q^{32} +(-4.77786 + 8.27550i) q^{33} +(0.356634 + 6.75453i) q^{34} +(-0.847502 + 0.176466i) q^{35} +(-15.3962 + 1.63036i) q^{36} +3.30212 q^{37} +(4.31250 + 4.40481i) q^{38} -14.0457i q^{39} +(5.91122 - 2.24890i) q^{40} +(1.84954 + 1.06783i) q^{41} +(0.0946096 + 1.79187i) q^{42} +(5.31922 - 9.21317i) q^{43} +(-5.32904 - 2.36766i) q^{44} +(11.5289 + 12.9116i) q^{45} +(-1.86609 + 3.66595i) q^{46} +(-5.15504 - 8.92879i) q^{47} +(-2.74563 - 12.8187i) q^{48} -6.85012 q^{49} +(-5.89962 - 3.89801i) q^{50} +(-7.83755 - 13.5750i) q^{51} +(8.52367 - 0.902603i) q^{52} +(0.978904 + 1.69551i) q^{53} +(18.4249 - 11.9755i) q^{54} +(1.32901 + 6.38274i) q^{55} +(-1.08132 + 0.172563i) q^{56} +(-13.5372 - 4.56342i) q^{57} +(-0.250715 - 0.127622i) q^{58} +(3.36162 - 5.82250i) q^{59} +(-9.84402 + 10.8590i) q^{60} +(2.73003 + 4.72855i) q^{61} +(-4.37560 - 6.73207i) q^{62} +(-1.49847 - 2.59542i) q^{63} +(7.60264 - 2.48995i) q^{64} +(-6.38266 - 7.14816i) q^{65} +(13.4950 - 0.712528i) q^{66} +(-4.41226 + 2.54742i) q^{67} +(7.73439 - 5.62859i) q^{68} -9.53300i q^{69} +(0.862415 + 0.868931i) q^{70} +(-2.23389 + 3.86921i) q^{71} +(13.7990 + 16.9997i) q^{72} +(9.16103 + 5.28912i) q^{73} +(-2.54494 - 3.91552i) q^{74} +(16.2823 + 1.84822i) q^{75} +(1.89939 - 8.50837i) q^{76} -1.12878i q^{77} +(-16.6548 + 10.8250i) q^{78} +(0.441518 - 0.764732i) q^{79} +(-7.22242 - 5.27605i) q^{80} +(-13.8509 + 23.9905i) q^{81} +(-0.159247 - 3.01608i) q^{82} -5.68186 q^{83} +(2.05182 - 1.49318i) q^{84} +(-10.1575 - 3.34708i) q^{85} +(-15.0241 + 0.793261i) q^{86} +0.651964 q^{87} +(1.29962 + 8.14371i) q^{88} +(-14.2851 + 8.24748i) q^{89} +(6.42474 - 23.6215i) q^{90} +(0.829583 + 1.43688i) q^{91} +(5.78513 - 0.612608i) q^{92} +(16.1142 + 9.30353i) q^{93} +(-6.61440 + 12.9940i) q^{94} +(-8.96311 + 3.82919i) q^{95} +(-13.0839 + 13.1350i) q^{96} +(0.281138 - 0.486945i) q^{97} +(5.27938 + 8.12259i) q^{98} +(-19.5467 + 11.2853i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 112 q - 2 q^{5} - 8 q^{6} + 44 q^{9} + 6 q^{10} - 36 q^{14} - 4 q^{16} + 44 q^{20} - 48 q^{21} + 2 q^{24} - 2 q^{25} - 36 q^{26} - 12 q^{29} - 32 q^{30} - 30 q^{34} + 20 q^{36} - 24 q^{40} - 24 q^{41} - 14 q^{44}+ \cdots - 84 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.770699 − 1.18576i −0.544966 − 0.838458i
\(3\) 2.83828 + 1.63868i 1.63868 + 0.946095i 0.981287 + 0.192550i \(0.0616759\pi\)
0.657397 + 0.753544i \(0.271657\pi\)
\(4\) −0.812047 + 1.82773i −0.406023 + 0.913863i
\(5\) 2.18912 − 0.455815i 0.979003 − 0.203847i
\(6\) −0.244379 − 4.62845i −0.0997672 − 1.88956i
\(7\) −0.387143 −0.146326 −0.0731632 − 0.997320i \(-0.523309\pi\)
−0.0731632 + 0.997320i \(0.523309\pi\)
\(8\) 2.79308 − 0.445735i 0.987504 − 0.157591i
\(9\) 3.87057 + 6.70403i 1.29019 + 2.23468i
\(10\) −2.22764 − 2.24447i −0.704441 − 0.709763i
\(11\) 2.91567i 0.879108i 0.898216 + 0.439554i \(0.144863\pi\)
−0.898216 + 0.439554i \(0.855137\pi\)
\(12\) −5.29988 + 3.85692i −1.52994 + 1.11340i
\(13\) −2.14283 − 3.71149i −0.594315 − 1.02938i −0.993643 − 0.112575i \(-0.964090\pi\)
0.399329 − 0.916808i \(-0.369243\pi\)
\(14\) 0.298371 + 0.459059i 0.0797430 + 0.122689i
\(15\) 6.96027 + 2.29354i 1.79713 + 0.592189i
\(16\) −2.68116 − 2.96840i −0.670290 − 0.742099i
\(17\) −4.14205 − 2.39142i −1.00460 − 0.580003i −0.0949906 − 0.995478i \(-0.530282\pi\)
−0.909605 + 0.415475i \(0.863615\pi\)
\(18\) 4.96631 − 9.75635i 1.17057 − 2.29959i
\(19\) −4.27334 + 0.859404i −0.980371 + 0.197161i
\(20\) −0.944559 + 4.37125i −0.211210 + 0.977441i
\(21\) −1.09882 − 0.634406i −0.239783 − 0.138439i
\(22\) 3.45728 − 2.24710i 0.737095 − 0.479084i
\(23\) −1.45437 − 2.51904i −0.303257 − 0.525256i 0.673615 − 0.739082i \(-0.264741\pi\)
−0.976872 + 0.213826i \(0.931407\pi\)
\(24\) 8.65799 + 3.31186i 1.76730 + 0.676031i
\(25\) 4.58446 − 1.99567i 0.916893 − 0.399133i
\(26\) −2.74946 + 5.40132i −0.539213 + 1.05929i
\(27\) 15.5385i 2.99038i
\(28\) 0.314378 − 0.707592i 0.0594119 − 0.133722i
\(29\) 0.172278 − 0.0994646i 0.0319912 − 0.0184701i −0.483919 − 0.875113i \(-0.660787\pi\)
0.515910 + 0.856643i \(0.327454\pi\)
\(30\) −2.64469 − 10.0208i −0.482853 − 1.82954i
\(31\) 5.67744 1.01970 0.509849 − 0.860264i \(-0.329701\pi\)
0.509849 + 0.860264i \(0.329701\pi\)
\(32\) −1.45343 + 5.46695i −0.256933 + 0.966429i
\(33\) −4.77786 + 8.27550i −0.831719 + 1.44058i
\(34\) 0.356634 + 6.75453i 0.0611623 + 1.15839i
\(35\) −0.847502 + 0.176466i −0.143254 + 0.0298282i
\(36\) −15.3962 + 1.63036i −2.56603 + 0.271727i
\(37\) 3.30212 0.542866 0.271433 − 0.962457i \(-0.412502\pi\)
0.271433 + 0.962457i \(0.412502\pi\)
\(38\) 4.31250 + 4.40481i 0.699580 + 0.714554i
\(39\) − 14.0457i − 2.24911i
\(40\) 5.91122 − 2.24890i 0.934645 − 0.355582i
\(41\) 1.84954 + 1.06783i 0.288849 + 0.166767i 0.637423 − 0.770514i \(-0.280000\pi\)
−0.348573 + 0.937281i \(0.613334\pi\)
\(42\) 0.0946096 + 1.79187i 0.0145986 + 0.276492i
\(43\) 5.31922 − 9.21317i 0.811174 − 1.40499i −0.100869 − 0.994900i \(-0.532162\pi\)
0.912043 − 0.410095i \(-0.134504\pi\)
\(44\) −5.32904 − 2.36766i −0.803384 − 0.356938i
\(45\) 11.5289 + 12.9116i 1.71863 + 1.92475i
\(46\) −1.86609 + 3.66595i −0.275140 + 0.540515i
\(47\) −5.15504 − 8.92879i −0.751939 − 1.30240i −0.946882 − 0.321583i \(-0.895785\pi\)
0.194942 − 0.980815i \(-0.437548\pi\)
\(48\) −2.74563 − 12.8187i −0.396298 − 1.85022i
\(49\) −6.85012 −0.978589
\(50\) −5.89962 − 3.89801i −0.834332 − 0.551262i
\(51\) −7.83755 − 13.5750i −1.09748 − 1.90088i
\(52\) 8.52367 − 0.902603i 1.18202 − 0.125168i
\(53\) 0.978904 + 1.69551i 0.134463 + 0.232896i 0.925392 − 0.379011i \(-0.123736\pi\)
−0.790929 + 0.611907i \(0.790402\pi\)
\(54\) 18.4249 − 11.9755i 2.50731 − 1.62966i
\(55\) 1.32901 + 6.38274i 0.179203 + 0.860649i
\(56\) −1.08132 + 0.172563i −0.144498 + 0.0230597i
\(57\) −13.5372 − 4.56342i −1.79305 − 0.604439i
\(58\) −0.250715 − 0.127622i −0.0329205 − 0.0167577i
\(59\) 3.36162 − 5.82250i 0.437646 − 0.758025i −0.559862 − 0.828586i \(-0.689146\pi\)
0.997507 + 0.0705615i \(0.0224791\pi\)
\(60\) −9.84402 + 10.8590i −1.27086 + 1.40189i
\(61\) 2.73003 + 4.72855i 0.349545 + 0.605429i 0.986169 − 0.165745i \(-0.0530029\pi\)
−0.636624 + 0.771175i \(0.719670\pi\)
\(62\) −4.37560 − 6.73207i −0.555701 − 0.854974i
\(63\) −1.49847 − 2.59542i −0.188789 − 0.326992i
\(64\) 7.60264 − 2.48995i 0.950330 − 0.311244i
\(65\) −6.38266 − 7.14816i −0.791672 − 0.886620i
\(66\) 13.4950 − 0.712528i 1.66112 − 0.0877061i
\(67\) −4.41226 + 2.54742i −0.539043 + 0.311217i −0.744691 − 0.667409i \(-0.767403\pi\)
0.205648 + 0.978626i \(0.434070\pi\)
\(68\) 7.73439 − 5.62859i 0.937933 − 0.682567i
\(69\) − 9.53300i − 1.14764i
\(70\) 0.862415 + 0.868931i 0.103078 + 0.103857i
\(71\) −2.23389 + 3.86921i −0.265114 + 0.459190i −0.967593 − 0.252513i \(-0.918743\pi\)
0.702480 + 0.711704i \(0.252076\pi\)
\(72\) 13.7990 + 16.9997i 1.62623 + 2.00343i
\(73\) 9.16103 + 5.28912i 1.07222 + 0.619045i 0.928786 − 0.370615i \(-0.120853\pi\)
0.143431 + 0.989660i \(0.454187\pi\)
\(74\) −2.54494 − 3.91552i −0.295844 − 0.455170i
\(75\) 16.2823 + 1.84822i 1.88012 + 0.213414i
\(76\) 1.89939 − 8.50837i 0.217875 − 0.975977i
\(77\) − 1.12878i − 0.128637i
\(78\) −16.6548 + 10.8250i −1.88579 + 1.22569i
\(79\) 0.441518 − 0.764732i 0.0496747 − 0.0860390i −0.840119 − 0.542402i \(-0.817515\pi\)
0.889794 + 0.456363i \(0.150848\pi\)
\(80\) −7.22242 − 5.27605i −0.807491 − 0.589880i
\(81\) −13.8509 + 23.9905i −1.53899 + 2.66561i
\(82\) −0.159247 − 3.01608i −0.0175859 − 0.333071i
\(83\) −5.68186 −0.623665 −0.311833 − 0.950137i \(-0.600943\pi\)
−0.311833 + 0.950137i \(0.600943\pi\)
\(84\) 2.05182 − 1.49318i 0.223871 − 0.162919i
\(85\) −10.1575 − 3.34708i −1.10173 − 0.363041i
\(86\) −15.0241 + 0.793261i −1.62009 + 0.0855396i
\(87\) 0.651964 0.0698979
\(88\) 1.29962 + 8.14371i 0.138540 + 0.868123i
\(89\) −14.2851 + 8.24748i −1.51421 + 0.874231i −0.514351 + 0.857580i \(0.671967\pi\)
−0.999861 + 0.0166511i \(0.994700\pi\)
\(90\) 6.42474 − 23.6215i 0.677227 − 2.48993i
\(91\) 0.829583 + 1.43688i 0.0869639 + 0.150626i
\(92\) 5.78513 − 0.612608i 0.603141 − 0.0638688i
\(93\) 16.1142 + 9.30353i 1.67096 + 0.964731i
\(94\) −6.61440 + 12.9940i −0.682224 + 1.34023i
\(95\) −8.96311 + 3.82919i −0.919595 + 0.392867i
\(96\) −13.0839 + 13.1350i −1.33537 + 1.34059i
\(97\) 0.281138 − 0.486945i 0.0285452 − 0.0494418i −0.851400 − 0.524517i \(-0.824246\pi\)
0.879945 + 0.475075i \(0.157579\pi\)
\(98\) 5.27938 + 8.12259i 0.533298 + 0.820505i
\(99\) −19.5467 + 11.2853i −1.96452 + 1.13422i
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 380.2.s.a.179.19 yes 112
4.3 odd 2 inner 380.2.s.a.179.56 yes 112
5.4 even 2 inner 380.2.s.a.179.38 yes 112
19.12 odd 6 inner 380.2.s.a.259.1 yes 112
20.19 odd 2 inner 380.2.s.a.179.1 ✓ 112
76.31 even 6 inner 380.2.s.a.259.38 yes 112
95.69 odd 6 inner 380.2.s.a.259.56 yes 112
380.259 even 6 inner 380.2.s.a.259.19 yes 112
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.s.a.179.1 ✓ 112 20.19 odd 2 inner
380.2.s.a.179.19 yes 112 1.1 even 1 trivial
380.2.s.a.179.38 yes 112 5.4 even 2 inner
380.2.s.a.179.56 yes 112 4.3 odd 2 inner
380.2.s.a.259.1 yes 112 19.12 odd 6 inner
380.2.s.a.259.19 yes 112 380.259 even 6 inner
380.2.s.a.259.38 yes 112 76.31 even 6 inner
380.2.s.a.259.56 yes 112 95.69 odd 6 inner