Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1014,2,Mod(239,1014)] 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("1014.239"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1014, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.09683076496\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: 16.0.9349208943630483456.9
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 239.7
Root \(0.500000 + 2.74530i\) of defining polynomial
Character \(\chi\) \(=\) 1014.239
Dual form 1014.2.g.c.437.7

$q$-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+(0.707107 + 0.707107i) q^{2} +(0.796225 + 1.53819i) q^{3} +1.00000i q^{4} +(2.76293 + 2.76293i) q^{5} +(-0.524648 + 1.65068i) q^{6} +(-1.79623 - 1.79623i) q^{7} +(-0.707107 + 0.707107i) q^{8} +(-1.73205 + 2.44949i) q^{9} +3.90738i q^{10} +(0.412157 - 0.412157i) q^{11} +(-1.53819 + 0.796225i) q^{12} -2.54025i q^{14} +(-2.05000 + 6.44983i) q^{15} -1.00000 q^{16} +1.09400 q^{17} +(-2.95680 + 0.507306i) q^{18} +(-0.971553 + 0.971553i) q^{19} +(-2.76293 + 2.76293i) q^{20} +(1.33273 - 4.19313i) q^{21} +0.582877 q^{22} -1.75292 q^{23} +(-1.65068 - 0.524648i) q^{24} +10.2676i q^{25} +(-5.14688 - 0.713876i) q^{27} +(1.79623 - 1.79623i) q^{28} +5.92330i q^{29} +(-6.01029 + 3.11115i) q^{30} +(6.49983 - 6.49983i) q^{31} +(-0.707107 - 0.707107i) q^{32} +(0.962144 + 0.305805i) q^{33} +(0.773575 + 0.773575i) q^{34} -9.92570i q^{35} +(-2.44949 - 1.73205i) q^{36} +(2.18840 + 2.18840i) q^{37} -1.37398 q^{38} -3.90738 q^{40} +(-3.74650 - 3.74650i) q^{41} +(3.90738 - 2.02261i) q^{42} +3.76778i q^{43} +(0.412157 + 0.412157i) q^{44} +(-11.5533 + 1.98224i) q^{45} +(-1.23950 - 1.23950i) q^{46} +(5.51114 - 5.51114i) q^{47} +(-0.796225 - 1.53819i) q^{48} -0.547150i q^{49} +(-7.26029 + 7.26029i) q^{50} +(0.871071 + 1.68278i) q^{51} +3.04435i q^{53} +(-3.13461 - 4.14418i) q^{54} +2.27752 q^{55} +2.54025 q^{56} +(-2.26801 - 0.720857i) q^{57} +(-4.18840 + 4.18840i) q^{58} +(5.99556 - 5.99556i) q^{59} +(-6.44983 - 2.05000i) q^{60} +9.34533 q^{61} +9.19215 q^{62} +(7.51099 - 1.28868i) q^{63} -1.00000i q^{64} +(0.464102 + 0.896575i) q^{66} +(4.66788 - 4.66788i) q^{67} +1.09400i q^{68} +(-1.39572 - 2.69632i) q^{69} +(7.01853 - 7.01853i) q^{70} +(-0.601383 - 0.601383i) q^{71} +(-0.507306 - 2.95680i) q^{72} +(-5.18078 - 5.18078i) q^{73} +3.09487i q^{74} +(-15.7935 + 8.17533i) q^{75} +(-0.971553 - 0.971553i) q^{76} -1.48065 q^{77} -13.1089 q^{79} +(-2.76293 - 2.76293i) q^{80} +(-3.00000 - 8.48528i) q^{81} -5.29835i q^{82} +(5.15394 + 5.15394i) q^{83} +(4.19313 + 1.33273i) q^{84} +(3.02265 + 3.02265i) q^{85} +(-2.66422 + 2.66422i) q^{86} +(-9.11115 + 4.71628i) q^{87} +0.582877i q^{88} +(6.85191 - 6.85191i) q^{89} +(-9.57108 - 6.76778i) q^{90} -1.75292i q^{92} +(15.1733 + 4.82264i) q^{93} +7.79393 q^{94} -5.36867 q^{95} +(0.524648 - 1.65068i) q^{96} +(0.433704 - 0.433704i) q^{97} +(0.386893 - 0.386893i) q^{98} +(0.295697 + 1.72345i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{7} - 24 q^{15} - 16 q^{16} + 32 q^{19} - 24 q^{21} + 16 q^{28} + 16 q^{31} + 24 q^{33} + 24 q^{34} - 8 q^{37} - 48 q^{45} + 48 q^{55} - 24 q^{57} - 24 q^{58} - 24 q^{60} + 48 q^{61} - 48 q^{66}+ \cdots + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.707107 + 0.707107i 0.500000 + 0.500000i
\(3\) 0.796225 + 1.53819i 0.459701 + 0.888074i
\(4\) 1.00000i 0.500000i
\(5\) 2.76293 + 2.76293i 1.23562 + 1.23562i 0.961773 + 0.273849i \(0.0882968\pi\)
0.273849 + 0.961773i \(0.411703\pi\)
\(6\) −0.524648 + 1.65068i −0.214186 + 0.673887i
\(7\) −1.79623 − 1.79623i −0.678909 − 0.678909i 0.280844 − 0.959753i \(-0.409386\pi\)
−0.959753 + 0.280844i \(0.909386\pi\)
\(8\) −0.707107 + 0.707107i −0.250000 + 0.250000i
\(9\) −1.73205 + 2.44949i −0.577350 + 0.816497i
\(10\) 3.90738i 1.23562i
\(11\) 0.412157 − 0.412157i 0.124270 − 0.124270i −0.642237 − 0.766506i \(-0.721993\pi\)
0.766506 + 0.642237i \(0.221993\pi\)
\(12\) −1.53819 + 0.796225i −0.444037 + 0.229850i
\(13\) 0 0
\(14\) − 2.54025i − 0.678909i
\(15\) −2.05000 + 6.44983i −0.529307 + 1.66534i
\(16\) −1.00000 −0.250000
\(17\) 1.09400 0.265334 0.132667 − 0.991161i \(-0.457646\pi\)
0.132667 + 0.991161i \(0.457646\pi\)
\(18\) −2.95680 + 0.507306i −0.696923 + 0.119573i
\(19\) −0.971553 + 0.971553i −0.222890 + 0.222890i −0.809714 − 0.586825i \(-0.800378\pi\)
0.586825 + 0.809714i \(0.300378\pi\)
\(20\) −2.76293 + 2.76293i −0.617811 + 0.617811i
\(21\) 1.33273 − 4.19313i 0.290826 − 0.915017i
\(22\) 0.582877 0.124270
\(23\) −1.75292 −0.365509 −0.182755 − 0.983159i \(-0.558501\pi\)
−0.182755 + 0.983159i \(0.558501\pi\)
\(24\) −1.65068 − 0.524648i −0.336944 − 0.107093i
\(25\) 10.2676i 2.05352i
\(26\) 0 0
\(27\) −5.14688 − 0.713876i −0.990518 − 0.137386i
\(28\) 1.79623 − 1.79623i 0.339455 − 0.339455i
\(29\) 5.92330i 1.09993i 0.835188 + 0.549965i \(0.185359\pi\)
−0.835188 + 0.549965i \(0.814641\pi\)
\(30\) −6.01029 + 3.11115i −1.09732 + 0.568016i
\(31\) 6.49983 − 6.49983i 1.16740 − 1.16740i 0.184588 − 0.982816i \(-0.440905\pi\)
0.982816 − 0.184588i \(-0.0590950\pi\)
\(32\) −0.707107 − 0.707107i −0.125000 − 0.125000i
\(33\) 0.962144 + 0.305805i 0.167488 + 0.0532339i
\(34\) 0.773575 + 0.773575i 0.132667 + 0.132667i
\(35\) − 9.92570i − 1.67775i
\(36\) −2.44949 − 1.73205i −0.408248 − 0.288675i
\(37\) 2.18840 + 2.18840i 0.359772 + 0.359772i 0.863729 − 0.503957i \(-0.168123\pi\)
−0.503957 + 0.863729i \(0.668123\pi\)
\(38\) −1.37398 −0.222890
\(39\) 0 0
\(40\) −3.90738 −0.617811
\(41\) −3.74650 − 3.74650i −0.585105 − 0.585105i 0.351197 − 0.936302i \(-0.385775\pi\)
−0.936302 + 0.351197i \(0.885775\pi\)
\(42\) 3.90738 − 2.02261i 0.602922 − 0.312095i
\(43\) 3.76778i 0.574581i 0.957844 + 0.287290i \(0.0927545\pi\)
−0.957844 + 0.287290i \(0.907246\pi\)
\(44\) 0.412157 + 0.412157i 0.0621349 + 0.0621349i
\(45\) −11.5533 + 1.98224i −1.72227 + 0.295494i
\(46\) −1.23950 − 1.23950i −0.182755 − 0.182755i
\(47\) 5.51114 − 5.51114i 0.803883 − 0.803883i −0.179817 − 0.983700i \(-0.557551\pi\)
0.983700 + 0.179817i \(0.0575506\pi\)
\(48\) −0.796225 − 1.53819i −0.114925 − 0.222018i
\(49\) − 0.547150i − 0.0781643i
\(50\) −7.26029 + 7.26029i −1.02676 + 1.02676i
\(51\) 0.871071 + 1.68278i 0.121974 + 0.235636i
\(52\) 0 0
\(53\) 3.04435i 0.418173i 0.977897 + 0.209087i \(0.0670490\pi\)
−0.977897 + 0.209087i \(0.932951\pi\)
\(54\) −3.13461 − 4.14418i −0.426566 − 0.563952i
\(55\) 2.27752 0.307101
\(56\) 2.54025 0.339455
\(57\) −2.26801 − 0.720857i −0.300405 − 0.0954799i
\(58\) −4.18840 + 4.18840i −0.549965 + 0.549965i
\(59\) 5.99556 − 5.99556i 0.780556 − 0.780556i −0.199369 − 0.979925i \(-0.563889\pi\)
0.979925 + 0.199369i \(0.0638892\pi\)
\(60\) −6.44983 − 2.05000i −0.832670 − 0.264653i
\(61\) 9.34533 1.19655 0.598273 − 0.801292i \(-0.295854\pi\)
0.598273 + 0.801292i \(0.295854\pi\)
\(62\) 9.19215 1.16740
\(63\) 7.51099 − 1.28868i 0.946296 − 0.162359i
\(64\) − 1.00000i − 0.125000i
\(65\) 0 0
\(66\) 0.464102 + 0.896575i 0.0571270 + 0.110361i
\(67\) 4.66788 − 4.66788i 0.570272 − 0.570272i −0.361932 − 0.932204i \(-0.617883\pi\)
0.932204 + 0.361932i \(0.117883\pi\)
\(68\) 1.09400i 0.132667i
\(69\) −1.39572 − 2.69632i −0.168025 − 0.324599i
\(70\) 7.01853 − 7.01853i 0.838875 − 0.838875i
\(71\) −0.601383 − 0.601383i −0.0713711 − 0.0713711i 0.670520 − 0.741891i \(-0.266071\pi\)
−0.741891 + 0.670520i \(0.766071\pi\)
\(72\) −0.507306 − 2.95680i −0.0597866 − 0.348462i
\(73\) −5.18078 − 5.18078i −0.606365 − 0.606365i 0.335629 − 0.941994i \(-0.391051\pi\)
−0.941994 + 0.335629i \(0.891051\pi\)
\(74\) 3.09487i 0.359772i
\(75\) −15.7935 + 8.17533i −1.82368 + 0.944006i
\(76\) −0.971553 − 0.971553i −0.111445 − 0.111445i
\(77\) −1.48065 −0.168736
\(78\) 0 0
\(79\) −13.1089 −1.47486 −0.737431 − 0.675422i \(-0.763961\pi\)
−0.737431 + 0.675422i \(0.763961\pi\)
\(80\) −2.76293 − 2.76293i −0.308905 − 0.308905i
\(81\) −3.00000 − 8.48528i −0.333333 − 0.942809i
\(82\) − 5.29835i − 0.585105i
\(83\) 5.15394 + 5.15394i 0.565719 + 0.565719i 0.930926 − 0.365208i \(-0.119002\pi\)
−0.365208 + 0.930926i \(0.619002\pi\)
\(84\) 4.19313 + 1.33273i 0.457508 + 0.145413i
\(85\) 3.02265 + 3.02265i 0.327852 + 0.327852i
\(86\) −2.66422 + 2.66422i −0.287290 + 0.287290i
\(87\) −9.11115 + 4.71628i −0.976818 + 0.505638i
\(88\) 0.582877i 0.0621349i
\(89\) 6.85191 − 6.85191i 0.726301 − 0.726301i −0.243580 − 0.969881i \(-0.578322\pi\)
0.969881 + 0.243580i \(0.0783219\pi\)
\(90\) −9.57108 − 6.76778i −1.00888 − 0.713386i
\(91\) 0 0
\(92\) − 1.75292i − 0.182755i
\(93\) 15.1733 + 4.82264i 1.57340 + 0.500084i
\(94\) 7.79393 0.803883
\(95\) −5.36867 −0.550814
\(96\) 0.524648 − 1.65068i 0.0535466 − 0.168472i
\(97\) 0.433704 − 0.433704i 0.0440360 − 0.0440360i −0.684746 − 0.728782i \(-0.740087\pi\)
0.728782 + 0.684746i \(0.240087\pi\)
\(98\) 0.386893 − 0.386893i 0.0390821 − 0.0390821i
\(99\) 0.295697 + 1.72345i 0.0297187 + 0.173213i
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 1014.2.g.c.239.7 16
3.2 odd 2 inner 1014.2.g.c.239.3 16
13.3 even 3 78.2.k.a.71.1 yes 16
13.5 odd 4 1014.2.g.d.437.7 16
13.7 odd 12 78.2.k.a.11.4 yes 16
13.8 odd 4 inner 1014.2.g.c.437.3 16
13.12 even 2 1014.2.g.d.239.3 16
39.5 even 4 1014.2.g.d.437.3 16
39.8 even 4 inner 1014.2.g.c.437.7 16
39.20 even 12 78.2.k.a.11.1 ✓ 16
39.29 odd 6 78.2.k.a.71.4 yes 16
39.38 odd 2 1014.2.g.d.239.7 16
52.3 odd 6 624.2.cn.d.305.4 16
52.7 even 12 624.2.cn.d.401.2 16
156.59 odd 12 624.2.cn.d.401.4 16
156.107 even 6 624.2.cn.d.305.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.k.a.11.1 ✓ 16 39.20 even 12
78.2.k.a.11.4 yes 16 13.7 odd 12
78.2.k.a.71.1 yes 16 13.3 even 3
78.2.k.a.71.4 yes 16 39.29 odd 6
624.2.cn.d.305.2 16 156.107 even 6
624.2.cn.d.305.4 16 52.3 odd 6
624.2.cn.d.401.2 16 52.7 even 12
624.2.cn.d.401.4 16 156.59 odd 12
1014.2.g.c.239.3 16 3.2 odd 2 inner
1014.2.g.c.239.7 16 1.1 even 1 trivial
1014.2.g.c.437.3 16 13.8 odd 4 inner
1014.2.g.c.437.7 16 39.8 even 4 inner
1014.2.g.d.239.3 16 13.12 even 2
1014.2.g.d.239.7 16 39.38 odd 2
1014.2.g.d.437.3 16 39.5 even 4
1014.2.g.d.437.7 16 13.5 odd 4