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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [234,2,Mod(41,234)] 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("234.41"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(234, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([10, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 234.z (of order \(12\), degree \(4\), 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: \(1.86849940730\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 227.7
Character \(\chi\) \(=\) 234.227
Dual form 234.2.z.a.167.7

$q$-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+(-0.258819 + 0.965926i) q^{2} +(1.73204 + 0.00498176i) q^{3} +(-0.866025 - 0.500000i) q^{4} +(0.361873 - 1.35053i) q^{5} +(-0.453098 + 1.67174i) q^{6} +(-0.715284 - 0.715284i) q^{7} +(0.707107 - 0.707107i) q^{8} +(2.99995 + 0.0172572i) q^{9} +(1.21085 + 0.699086i) q^{10} +(5.61718 + 1.50512i) q^{11} +(-1.49750 - 0.870336i) q^{12} +(-0.218517 + 3.59892i) q^{13} +(0.876041 - 0.505782i) q^{14} +(0.633509 - 2.33737i) q^{15} +(0.500000 + 0.866025i) q^{16} +(-1.67305 - 2.89781i) q^{17} +(-0.793114 + 2.89326i) q^{18} +(-7.07605 - 1.89602i) q^{19} +(-0.988657 + 0.988657i) q^{20} +(-1.23534 - 1.24247i) q^{21} +(-2.90767 + 5.03622i) q^{22} +0.580436 q^{23} +(1.22826 - 1.22122i) q^{24} +(2.63715 + 1.52256i) q^{25} +(-3.41974 - 1.14254i) q^{26} +(5.19596 + 0.0448353i) q^{27} +(0.261812 + 0.977097i) q^{28} +(-0.892249 + 0.515140i) q^{29} +(2.09377 + 1.21688i) q^{30} +(-8.52714 - 2.28484i) q^{31} +(-0.965926 + 0.258819i) q^{32} +(9.72170 + 2.63491i) q^{33} +(3.23209 - 0.866036i) q^{34} +(-1.22486 + 0.707171i) q^{35} +(-2.58940 - 1.51492i) q^{36} +(-7.82443 + 2.09655i) q^{37} +(3.66283 - 6.34422i) q^{38} +(-0.396411 + 6.23240i) q^{39} +(-0.699086 - 1.21085i) q^{40} +(1.15731 + 1.15731i) q^{41} +(1.51986 - 0.871673i) q^{42} +8.38054i q^{43} +(-4.11206 - 4.11206i) q^{44} +(1.10891 - 4.04528i) q^{45} +(-0.150228 + 0.560658i) q^{46} +(-0.388741 - 1.45080i) q^{47} +(0.861707 + 1.50248i) q^{48} -5.97674i q^{49} +(-2.15322 + 2.15322i) q^{50} +(-2.88336 - 5.02747i) q^{51} +(1.98870 - 3.00750i) q^{52} -2.77082i q^{53} +(-1.38812 + 5.00731i) q^{54} +(4.06542 - 7.04151i) q^{55} -1.01156 q^{56} +(-12.2466 - 3.31924i) q^{57} +(-0.266656 - 0.995175i) q^{58} +(-3.10182 - 11.5761i) q^{59} +(-1.71732 + 1.70747i) q^{60} +2.82816 q^{61} +(4.41397 - 7.64523i) q^{62} +(-2.13347 - 2.15816i) q^{63} -1.00000i q^{64} +(4.78138 + 1.59747i) q^{65} +(-5.06129 + 8.70847i) q^{66} +(-0.802042 + 0.802042i) q^{67} +3.34611i q^{68} +(1.00534 + 0.00289159i) q^{69} +(-0.366058 - 1.36615i) q^{70} +(-2.71495 + 10.1323i) q^{71} +(2.13349 - 2.10908i) q^{72} +(0.788181 + 0.788181i) q^{73} -8.10045i q^{74} +(4.56007 + 2.65027i) q^{75} +(5.18003 + 5.18003i) q^{76} +(-2.94129 - 5.09447i) q^{77} +(-5.91744 - 1.99597i) q^{78} +(-0.827245 + 1.43283i) q^{79} +(1.35053 - 0.361873i) q^{80} +(8.99940 + 0.103542i) q^{81} +(-1.41741 + 0.818343i) q^{82} +(-15.7224 + 4.21280i) q^{83} +(0.448603 + 1.69368i) q^{84} +(-4.51902 + 1.21087i) q^{85} +(-8.09498 - 2.16904i) q^{86} +(-1.54798 + 0.887801i) q^{87} +(5.03622 - 2.90767i) q^{88} +(0.783797 + 2.92517i) q^{89} +(3.62043 + 2.11812i) q^{90} +(2.73056 - 2.41795i) q^{91} +(-0.502672 - 0.290218i) q^{92} +(-14.7580 - 3.99992i) q^{93} +1.50198 q^{94} +(-5.12127 + 8.87030i) q^{95} +(-1.67432 + 0.443474i) q^{96} +(8.70487 - 8.70487i) q^{97} +(5.77308 + 1.54689i) q^{98} +(16.8253 + 4.61222i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 8 q^{7} - 24 q^{11} + 28 q^{16} - 8 q^{19} - 24 q^{21} + 36 q^{27} - 4 q^{28} - 24 q^{30} - 4 q^{31} + 60 q^{33} - 24 q^{35} - 4 q^{37} + 36 q^{38} - 48 q^{41} - 36 q^{42} - 84 q^{45} - 36 q^{47}+ \cdots + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/234\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\)
\(\chi(n)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.258819 + 0.965926i −0.183013 + 0.683013i
\(3\) 1.73204 + 0.00498176i 0.999996 + 0.00287622i
\(4\) −0.866025 − 0.500000i −0.433013 − 0.250000i
\(5\) 0.361873 − 1.35053i 0.161835 − 0.603975i −0.836588 − 0.547833i \(-0.815453\pi\)
0.998423 − 0.0561429i \(-0.0178802\pi\)
\(6\) −0.453098 + 1.67174i −0.184976 + 0.682483i
\(7\) −0.715284 − 0.715284i −0.270352 − 0.270352i 0.558890 − 0.829242i \(-0.311227\pi\)
−0.829242 + 0.558890i \(0.811227\pi\)
\(8\) 0.707107 − 0.707107i 0.250000 − 0.250000i
\(9\) 2.99995 + 0.0172572i 0.999983 + 0.00575242i
\(10\) 1.21085 + 0.699086i 0.382905 + 0.221070i
\(11\) 5.61718 + 1.50512i 1.69364 + 0.453810i 0.971326 − 0.237752i \(-0.0764106\pi\)
0.722317 + 0.691562i \(0.243077\pi\)
\(12\) −1.49750 − 0.870336i −0.432292 − 0.251244i
\(13\) −0.218517 + 3.59892i −0.0606058 + 0.998162i
\(14\) 0.876041 − 0.505782i 0.234132 − 0.135176i
\(15\) 0.633509 − 2.33737i 0.163571 − 0.603507i
\(16\) 0.500000 + 0.866025i 0.125000 + 0.216506i
\(17\) −1.67305 − 2.89781i −0.405775 − 0.702823i 0.588636 − 0.808398i \(-0.299665\pi\)
−0.994411 + 0.105575i \(0.966332\pi\)
\(18\) −0.793114 + 2.89326i −0.186939 + 0.681949i
\(19\) −7.07605 − 1.89602i −1.62336 − 0.434977i −0.671372 − 0.741120i \(-0.734295\pi\)
−0.951986 + 0.306143i \(0.900961\pi\)
\(20\) −0.988657 + 0.988657i −0.221070 + 0.221070i
\(21\) −1.23534 − 1.24247i −0.269573 − 0.271129i
\(22\) −2.90767 + 5.03622i −0.619916 + 1.07373i
\(23\) 0.580436 0.121029 0.0605146 − 0.998167i \(-0.480726\pi\)
0.0605146 + 0.998167i \(0.480726\pi\)
\(24\) 1.22826 − 1.22122i 0.250718 − 0.249280i
\(25\) 2.63715 + 1.52256i 0.527430 + 0.304512i
\(26\) −3.41974 − 1.14254i −0.670666 − 0.224071i
\(27\) 5.19596 + 0.0448353i 0.999963 + 0.00862856i
\(28\) 0.261812 + 0.977097i 0.0494779 + 0.184654i
\(29\) −0.892249 + 0.515140i −0.165687 + 0.0956592i −0.580550 − 0.814224i \(-0.697163\pi\)
0.414864 + 0.909884i \(0.363829\pi\)
\(30\) 2.09377 + 1.21688i 0.382268 + 0.222171i
\(31\) −8.52714 − 2.28484i −1.53152 − 0.410369i −0.608005 − 0.793933i \(-0.708030\pi\)
−0.923514 + 0.383564i \(0.874697\pi\)
\(32\) −0.965926 + 0.258819i −0.170753 + 0.0457532i
\(33\) 9.72170 + 2.63491i 1.69233 + 0.458680i
\(34\) 3.23209 − 0.866036i 0.554299 − 0.148524i
\(35\) −1.22486 + 0.707171i −0.207038 + 0.119534i
\(36\) −2.58940 − 1.51492i −0.431567 − 0.252487i
\(37\) −7.82443 + 2.09655i −1.28633 + 0.344671i −0.836265 − 0.548326i \(-0.815265\pi\)
−0.450064 + 0.892996i \(0.648599\pi\)
\(38\) 3.66283 − 6.34422i 0.594190 − 1.02917i
\(39\) −0.396411 + 6.23240i −0.0634765 + 0.997983i
\(40\) −0.699086 − 1.21085i −0.110535 − 0.191453i
\(41\) 1.15731 + 1.15731i 0.180742 + 0.180742i 0.791679 − 0.610937i \(-0.209207\pi\)
−0.610937 + 0.791679i \(0.709207\pi\)
\(42\) 1.51986 − 0.871673i 0.234520 − 0.134502i
\(43\) 8.38054i 1.27802i 0.769198 + 0.639010i \(0.220656\pi\)
−0.769198 + 0.639010i \(0.779344\pi\)
\(44\) −4.11206 − 4.11206i −0.619916 − 0.619916i
\(45\) 1.10891 − 4.04528i 0.165306 − 0.603035i
\(46\) −0.150228 + 0.560658i −0.0221499 + 0.0826645i
\(47\) −0.388741 − 1.45080i −0.0567037 − 0.211621i 0.931761 − 0.363072i \(-0.118272\pi\)
−0.988465 + 0.151451i \(0.951605\pi\)
\(48\) 0.861707 + 1.50248i 0.124377 + 0.216865i
\(49\) − 5.97674i − 0.853820i
\(50\) −2.15322 + 2.15322i −0.304512 + 0.304512i
\(51\) −2.88336 − 5.02747i −0.403752 − 0.703987i
\(52\) 1.98870 − 3.00750i 0.275784 − 0.417065i
\(53\) − 2.77082i − 0.380601i −0.981726 − 0.190300i \(-0.939054\pi\)
0.981726 − 0.190300i \(-0.0609462\pi\)
\(54\) −1.38812 + 5.00731i −0.188899 + 0.681408i
\(55\) 4.06542 − 7.04151i 0.548180 − 0.949476i
\(56\) −1.01156 −0.135176
\(57\) −12.2466 − 3.31924i −1.62210 − 0.439645i
\(58\) −0.266656 − 0.995175i −0.0350137 − 0.130673i
\(59\) −3.10182 − 11.5761i −0.403822 − 1.50708i −0.806218 − 0.591619i \(-0.798489\pi\)
0.402395 − 0.915466i \(-0.368178\pi\)
\(60\) −1.71732 + 1.70747i −0.221705 + 0.220434i
\(61\) 2.82816 0.362108 0.181054 − 0.983473i \(-0.442049\pi\)
0.181054 + 0.983473i \(0.442049\pi\)
\(62\) 4.41397 − 7.64523i 0.560575 − 0.970945i
\(63\) −2.13347 − 2.15816i −0.268792 − 0.271903i
\(64\) − 1.00000i − 0.125000i
\(65\) 4.78138 + 1.59747i 0.593057 + 0.198142i
\(66\) −5.06129 + 8.70847i −0.623002 + 1.07194i
\(67\) −0.802042 + 0.802042i −0.0979851 + 0.0979851i −0.754400 − 0.656415i \(-0.772072\pi\)
0.656415 + 0.754400i \(0.272072\pi\)
\(68\) 3.34611i 0.405775i
\(69\) 1.00534 + 0.00289159i 0.121029 + 0.000348107i
\(70\) −0.366058 − 1.36615i −0.0437524 − 0.163286i
\(71\) −2.71495 + 10.1323i −0.322205 + 1.20249i 0.594886 + 0.803810i \(0.297197\pi\)
−0.917091 + 0.398677i \(0.869470\pi\)
\(72\) 2.13349 − 2.10908i 0.251434 − 0.248558i
\(73\) 0.788181 + 0.788181i 0.0922496 + 0.0922496i 0.751726 − 0.659476i \(-0.229222\pi\)
−0.659476 + 0.751726i \(0.729222\pi\)
\(74\) − 8.10045i − 0.941658i
\(75\) 4.56007 + 2.65027i 0.526551 + 0.306027i
\(76\) 5.18003 + 5.18003i 0.594190 + 0.594190i
\(77\) −2.94129 − 5.09447i −0.335191 − 0.580568i
\(78\) −5.91744 − 1.99597i −0.670018 − 0.225999i
\(79\) −0.827245 + 1.43283i −0.0930723 + 0.161206i −0.908802 − 0.417227i \(-0.863002\pi\)
0.815730 + 0.578433i \(0.196335\pi\)
\(80\) 1.35053 − 0.361873i 0.150994 − 0.0404587i
\(81\) 8.99940 + 0.103542i 0.999934 + 0.0115046i
\(82\) −1.41741 + 0.818343i −0.156527 + 0.0903709i
\(83\) −15.7224 + 4.21280i −1.72576 + 0.462415i −0.979198 − 0.202906i \(-0.934961\pi\)
−0.746557 + 0.665321i \(0.768295\pi\)
\(84\) 0.448603 + 1.69368i 0.0489466 + 0.184795i
\(85\) −4.51902 + 1.21087i −0.490156 + 0.131337i
\(86\) −8.09498 − 2.16904i −0.872904 − 0.233894i
\(87\) −1.54798 + 0.887801i −0.165961 + 0.0951822i
\(88\) 5.03622 − 2.90767i 0.536863 − 0.309958i
\(89\) 0.783797 + 2.92517i 0.0830824 + 0.310068i 0.994944 − 0.100430i \(-0.0320218\pi\)
−0.911862 + 0.410497i \(0.865355\pi\)
\(90\) 3.62043 + 2.11812i 0.381627 + 0.223269i
\(91\) 2.73056 − 2.41795i 0.286240 − 0.253470i
\(92\) −0.502672 − 0.290218i −0.0524072 − 0.0302573i
\(93\) −14.7580 − 3.99992i −1.53033 − 0.414773i
\(94\) 1.50198 0.154917
\(95\) −5.12127 + 8.87030i −0.525431 + 0.910074i
\(96\) −1.67432 + 0.443474i −0.170884 + 0.0452619i
\(97\) 8.70487 − 8.70487i 0.883846 − 0.883846i −0.110077 − 0.993923i \(-0.535110\pi\)
0.993923 + 0.110077i \(0.0351099\pi\)
\(98\) 5.77308 + 1.54689i 0.583170 + 0.156260i
\(99\) 16.8253 + 4.61222i 1.69100 + 0.463545i
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 234.2.z.a.227.7 yes 56
3.2 odd 2 702.2.bc.a.305.10 56
9.4 even 3 702.2.bb.a.71.3 56
9.5 odd 6 234.2.y.a.149.10 yes 56
13.11 odd 12 234.2.y.a.11.10 ✓ 56
39.11 even 12 702.2.bb.a.89.3 56
117.50 even 12 inner 234.2.z.a.167.7 yes 56
117.76 odd 12 702.2.bc.a.557.10 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
234.2.y.a.11.10 ✓ 56 13.11 odd 12
234.2.y.a.149.10 yes 56 9.5 odd 6
234.2.z.a.167.7 yes 56 117.50 even 12 inner
234.2.z.a.227.7 yes 56 1.1 even 1 trivial
702.2.bb.a.71.3 56 9.4 even 3
702.2.bb.a.89.3 56 39.11 even 12
702.2.bc.a.305.10 56 3.2 odd 2
702.2.bc.a.557.10 56 117.76 odd 12