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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [171,2,Mod(106,171)] 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("171.106"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 121.16
Character \(\chi\) \(=\) 171.121
Dual form 171.2.g.c.106.16

$q$-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+(1.30515 - 2.26059i) q^{2} +(1.63157 + 0.581354i) q^{3} +(-2.40684 - 4.16877i) q^{4} -2.00201 q^{5} +(3.44365 - 2.92956i) q^{6} +(0.257107 + 0.445323i) q^{7} -7.34455 q^{8} +(2.32405 + 1.89704i) q^{9} +(-2.61292 + 4.52572i) q^{10} +(2.04883 + 3.54868i) q^{11} +(-1.50340 - 8.20087i) q^{12} +(-1.85122 - 3.20641i) q^{13} +1.34226 q^{14} +(-3.26642 - 1.16388i) q^{15} +(-4.77208 + 8.26548i) q^{16} +(3.60664 + 6.24688i) q^{17} +(7.32168 - 2.77780i) q^{18} +(0.559954 - 4.32278i) q^{19} +(4.81851 + 8.34591i) q^{20} +(0.160599 + 0.876047i) q^{21} +10.6961 q^{22} +(-0.174335 - 0.301956i) q^{23} +(-11.9832 - 4.26979i) q^{24} -0.991962 q^{25} -9.66450 q^{26} +(2.68901 + 4.44626i) q^{27} +(1.23763 - 2.14364i) q^{28} -7.54782 q^{29} +(-6.89422 + 5.86500i) q^{30} +(-0.773246 + 1.33930i) q^{31} +(5.11201 + 8.85425i) q^{32} +(1.27977 + 6.98102i) q^{33} +18.8288 q^{34} +(-0.514731 - 0.891541i) q^{35} +(2.31471 - 14.2543i) q^{36} -6.82195 q^{37} +(-9.04121 - 6.90771i) q^{38} +(-1.15634 - 6.30770i) q^{39} +14.7039 q^{40} -2.92203 q^{41} +(2.18999 + 0.780327i) q^{42} +(-0.200878 + 0.347931i) q^{43} +(9.86241 - 17.0822i) q^{44} +(-4.65278 - 3.79790i) q^{45} -0.910132 q^{46} +6.16199 q^{47} +(-12.5912 + 10.7115i) q^{48} +(3.36779 - 5.83319i) q^{49} +(-1.29466 + 2.24242i) q^{50} +(2.25284 + 12.2890i) q^{51} +(-8.91119 + 15.4346i) q^{52} +(2.35955 - 4.08687i) q^{53} +(13.5607 - 0.275696i) q^{54} +(-4.10177 - 7.10448i) q^{55} +(-1.88834 - 3.27070i) q^{56} +(3.42667 - 6.72740i) q^{57} +(-9.85105 + 17.0625i) q^{58} -4.30674 q^{59} +(3.00982 + 16.4182i) q^{60} +10.9341 q^{61} +(2.01840 + 3.49598i) q^{62} +(-0.247266 + 1.52270i) q^{63} +7.59946 q^{64} +(3.70616 + 6.41926i) q^{65} +(17.4515 + 6.21824i) q^{66} +(-0.480007 - 0.831396i) q^{67} +(17.3612 - 30.0705i) q^{68} +(-0.108896 - 0.594013i) q^{69} -2.68721 q^{70} +(-3.26848 - 5.66117i) q^{71} +(-17.0691 - 13.9329i) q^{72} +(1.31999 + 2.28629i) q^{73} +(-8.90367 + 15.4216i) q^{74} +(-1.61846 - 0.576681i) q^{75} +(-19.3684 + 8.06993i) q^{76} +(-1.05354 + 1.82478i) q^{77} +(-15.7683 - 5.61850i) q^{78} +(-3.55860 + 6.16367i) q^{79} +(9.55374 - 16.5476i) q^{80} +(1.80246 + 8.81766i) q^{81} +(-3.81369 + 6.60551i) q^{82} +(-8.37625 - 14.5081i) q^{83} +(3.26550 - 2.77800i) q^{84} +(-7.22052 - 12.5063i) q^{85} +(0.524352 + 0.908205i) q^{86} +(-12.3148 - 4.38796i) q^{87} +(-15.0477 - 26.0634i) q^{88} +(-5.10443 + 8.84113i) q^{89} +(-14.6581 + 5.56118i) q^{90} +(0.951926 - 1.64878i) q^{91} +(-0.839190 + 1.45352i) q^{92} +(-2.04021 + 1.73564i) q^{93} +(8.04233 - 13.9297i) q^{94} +(-1.12103 + 8.65425i) q^{95} +(3.19315 + 17.4182i) q^{96} +(9.64219 - 16.7008i) q^{97} +(-8.79095 - 15.2264i) q^{98} +(-1.97040 + 12.1340i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + q^{2} - 2 q^{3} - 17 q^{4} - 6 q^{5} + 2 q^{6} + q^{7} - 36 q^{8} - 10 q^{9} - 8 q^{10} + 7 q^{11} - 3 q^{12} - 4 q^{13} - 2 q^{14} + q^{15} - 11 q^{16} - 7 q^{17} + 6 q^{18} + 7 q^{19} - 3 q^{20}+ \cdots - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.30515 − 2.26059i 0.922881 − 1.59848i 0.127948 − 0.991781i \(-0.459161\pi\)
0.794934 − 0.606696i \(-0.207506\pi\)
\(3\) 1.63157 + 0.581354i 0.941989 + 0.335645i
\(4\) −2.40684 − 4.16877i −1.20342 − 2.08438i
\(5\) −2.00201 −0.895325 −0.447663 − 0.894202i \(-0.647743\pi\)
−0.447663 + 0.894202i \(0.647743\pi\)
\(6\) 3.44365 − 2.92956i 1.40586 − 1.19599i
\(7\) 0.257107 + 0.445323i 0.0971775 + 0.168316i 0.910515 − 0.413475i \(-0.135685\pi\)
−0.813338 + 0.581792i \(0.802352\pi\)
\(8\) −7.34455 −2.59669
\(9\) 2.32405 + 1.89704i 0.774685 + 0.632348i
\(10\) −2.61292 + 4.52572i −0.826279 + 1.43116i
\(11\) 2.04883 + 3.54868i 0.617745 + 1.06997i 0.989896 + 0.141794i \(0.0452870\pi\)
−0.372151 + 0.928172i \(0.621380\pi\)
\(12\) −1.50340 − 8.20087i −0.433994 − 2.36739i
\(13\) −1.85122 − 3.20641i −0.513437 − 0.889298i −0.999879 − 0.0155853i \(-0.995039\pi\)
0.486442 − 0.873713i \(-0.338294\pi\)
\(14\) 1.34226 0.358733
\(15\) −3.26642 − 1.16388i −0.843386 − 0.300512i
\(16\) −4.77208 + 8.26548i −1.19302 + 2.06637i
\(17\) 3.60664 + 6.24688i 0.874738 + 1.51509i 0.857042 + 0.515247i \(0.172300\pi\)
0.0176966 + 0.999843i \(0.494367\pi\)
\(18\) 7.32168 − 2.77780i 1.72574 − 0.654734i
\(19\) 0.559954 − 4.32278i 0.128462 − 0.991714i
\(20\) 4.81851 + 8.34591i 1.07745 + 1.86620i
\(21\) 0.160599 + 0.876047i 0.0350455 + 0.191169i
\(22\) 10.6961 2.28042
\(23\) −0.174335 − 0.301956i −0.0363513 − 0.0629622i 0.847277 − 0.531151i \(-0.178240\pi\)
−0.883629 + 0.468188i \(0.844907\pi\)
\(24\) −11.9832 − 4.26979i −2.44605 − 0.871567i
\(25\) −0.991962 −0.198392
\(26\) −9.66450 −1.89536
\(27\) 2.68901 + 4.44626i 0.517500 + 0.855683i
\(28\) 1.23763 − 2.14364i 0.233891 − 0.405110i
\(29\) −7.54782 −1.40160 −0.700798 − 0.713360i \(-0.747172\pi\)
−0.700798 + 0.713360i \(0.747172\pi\)
\(30\) −6.89422 + 5.86500i −1.25871 + 1.07080i
\(31\) −0.773246 + 1.33930i −0.138879 + 0.240545i −0.927073 − 0.374882i \(-0.877683\pi\)
0.788194 + 0.615427i \(0.211017\pi\)
\(32\) 5.11201 + 8.85425i 0.903684 + 1.56523i
\(33\) 1.27977 + 6.98102i 0.222780 + 1.21524i
\(34\) 18.8288 3.22912
\(35\) −0.514731 − 0.891541i −0.0870055 − 0.150698i
\(36\) 2.31471 − 14.2543i 0.385785 − 2.37572i
\(37\) −6.82195 −1.12152 −0.560760 − 0.827978i \(-0.689491\pi\)
−0.560760 + 0.827978i \(0.689491\pi\)
\(38\) −9.04121 − 6.90771i −1.46668 − 1.12058i
\(39\) −1.15634 − 6.30770i −0.185163 − 1.01004i
\(40\) 14.7039 2.32488
\(41\) −2.92203 −0.456345 −0.228172 − 0.973621i \(-0.573275\pi\)
−0.228172 + 0.973621i \(0.573275\pi\)
\(42\) 2.18999 + 0.780327i 0.337922 + 0.120407i
\(43\) −0.200878 + 0.347931i −0.0306336 + 0.0530590i −0.880936 − 0.473236i \(-0.843086\pi\)
0.850302 + 0.526295i \(0.176419\pi\)
\(44\) 9.86241 − 17.0822i 1.48681 − 2.57524i
\(45\) −4.65278 − 3.79790i −0.693595 − 0.566157i
\(46\) −0.910132 −0.134192
\(47\) 6.16199 0.898819 0.449410 − 0.893326i \(-0.351634\pi\)
0.449410 + 0.893326i \(0.351634\pi\)
\(48\) −12.5912 + 10.7115i −1.81738 + 1.54607i
\(49\) 3.36779 − 5.83319i 0.481113 − 0.833312i
\(50\) −1.29466 + 2.24242i −0.183093 + 0.317126i
\(51\) 2.25284 + 12.2890i 0.315460 + 1.72080i
\(52\) −8.91119 + 15.4346i −1.23576 + 2.14040i
\(53\) 2.35955 − 4.08687i 0.324110 − 0.561374i −0.657222 − 0.753697i \(-0.728269\pi\)
0.981332 + 0.192323i \(0.0616020\pi\)
\(54\) 13.5607 − 0.275696i 1.84538 − 0.0375174i
\(55\) −4.10177 − 7.10448i −0.553083 − 0.957968i
\(56\) −1.88834 − 3.27070i −0.252340 − 0.437066i
\(57\) 3.42667 − 6.72740i 0.453874 − 0.891066i
\(58\) −9.85105 + 17.0625i −1.29351 + 2.24042i
\(59\) −4.30674 −0.560690 −0.280345 − 0.959899i \(-0.590449\pi\)
−0.280345 + 0.959899i \(0.590449\pi\)
\(60\) 3.00982 + 16.4182i 0.388566 + 2.11958i
\(61\) 10.9341 1.39997 0.699984 − 0.714159i \(-0.253191\pi\)
0.699984 + 0.714159i \(0.253191\pi\)
\(62\) 2.01840 + 3.49598i 0.256338 + 0.443990i
\(63\) −0.247266 + 1.52270i −0.0311525 + 0.191842i
\(64\) 7.59946 0.949933
\(65\) 3.70616 + 6.41926i 0.459693 + 0.796211i
\(66\) 17.4515 + 6.21824i 2.14813 + 0.765413i
\(67\) −0.480007 − 0.831396i −0.0586421 − 0.101571i 0.835214 − 0.549925i \(-0.185344\pi\)
−0.893856 + 0.448354i \(0.852010\pi\)
\(68\) 17.3612 − 30.0705i 2.10535 − 3.64658i
\(69\) −0.108896 − 0.594013i −0.0131095 − 0.0715108i
\(70\) −2.68721 −0.321183
\(71\) −3.26848 − 5.66117i −0.387897 − 0.671857i 0.604269 − 0.796780i \(-0.293465\pi\)
−0.992166 + 0.124923i \(0.960132\pi\)
\(72\) −17.0691 − 13.9329i −2.01162 − 1.64201i
\(73\) 1.31999 + 2.28629i 0.154493 + 0.267591i 0.932874 − 0.360202i \(-0.117292\pi\)
−0.778381 + 0.627792i \(0.783959\pi\)
\(74\) −8.90367 + 15.4216i −1.03503 + 1.79273i
\(75\) −1.61846 − 0.576681i −0.186883 − 0.0665894i
\(76\) −19.3684 + 8.06993i −2.22171 + 0.925684i
\(77\) −1.05354 + 1.82478i −0.120062 + 0.207953i
\(78\) −15.7683 − 5.61850i −1.78541 − 0.636170i
\(79\) −3.55860 + 6.16367i −0.400374 + 0.693467i −0.993771 − 0.111442i \(-0.964453\pi\)
0.593397 + 0.804910i \(0.297786\pi\)
\(80\) 9.55374 − 16.5476i 1.06814 − 1.85007i
\(81\) 1.80246 + 8.81766i 0.200273 + 0.979740i
\(82\) −3.81369 + 6.60551i −0.421152 + 0.729457i
\(83\) −8.37625 − 14.5081i −0.919413 − 1.59247i −0.800309 − 0.599587i \(-0.795331\pi\)
−0.119103 − 0.992882i \(-0.538002\pi\)
\(84\) 3.26550 − 2.77800i 0.356296 − 0.303105i
\(85\) −7.22052 − 12.5063i −0.783175 − 1.35650i
\(86\) 0.524352 + 0.908205i 0.0565424 + 0.0979343i
\(87\) −12.3148 − 4.38796i −1.32029 − 0.470439i
\(88\) −15.0477 − 26.0634i −1.60409 − 2.77837i
\(89\) −5.10443 + 8.84113i −0.541068 + 0.937158i 0.457775 + 0.889068i \(0.348647\pi\)
−0.998843 + 0.0480896i \(0.984687\pi\)
\(90\) −14.6581 + 5.56118i −1.54509 + 0.586200i
\(91\) 0.951926 − 1.64878i 0.0997889 − 0.172839i
\(92\) −0.839190 + 1.45352i −0.0874916 + 0.151540i
\(93\) −2.04021 + 1.73564i −0.211560 + 0.179977i
\(94\) 8.04233 − 13.9297i 0.829503 − 1.43674i
\(95\) −1.12103 + 8.65425i −0.115016 + 0.887907i
\(96\) 3.19315 + 17.4182i 0.325899 + 1.77774i
\(97\) 9.64219 − 16.7008i 0.979016 − 1.69571i 0.313028 − 0.949744i \(-0.398657\pi\)
0.665989 − 0.745962i \(-0.268010\pi\)
\(98\) −8.79095 − 15.2264i −0.888021 − 1.53810i
\(99\) −1.97040 + 12.1340i −0.198033 + 1.21952i
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 171.2.g.c.121.16 yes 32
3.2 odd 2 513.2.g.c.64.1 32
9.2 odd 6 513.2.h.c.235.16 32
9.7 even 3 171.2.h.c.7.1 yes 32
19.11 even 3 171.2.h.c.49.1 yes 32
57.11 odd 6 513.2.h.c.334.16 32
171.11 odd 6 513.2.g.c.505.1 32
171.106 even 3 inner 171.2.g.c.106.16 ✓ 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.g.c.106.16 ✓ 32 171.106 even 3 inner
171.2.g.c.121.16 yes 32 1.1 even 1 trivial
171.2.h.c.7.1 yes 32 9.7 even 3
171.2.h.c.49.1 yes 32 19.11 even 3
513.2.g.c.64.1 32 3.2 odd 2
513.2.g.c.505.1 32 171.11 odd 6
513.2.h.c.235.16 32 9.2 odd 6
513.2.h.c.334.16 32 57.11 odd 6