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 437.3
Root \(0.500000 + 1.74530i\) of defining polynomial
Character \(\chi\) \(=\) 1014.437
Dual form 1014.2.g.c.239.3

$q$-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} +(-1.65068 - 0.524648i) 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} +(-6.44983 - 2.05000i) q^{15} -1.00000 q^{16} -1.09400 q^{17} +(-0.507306 - 2.95680i) q^{18} +(-0.971553 - 0.971553i) q^{19} +(2.76293 + 2.76293i) q^{20} +(-4.19313 - 1.33273i) q^{21} +0.582877 q^{22} +1.75292 q^{23} +(-0.524648 + 1.65068i) 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} +(0.305805 - 0.962144i) 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} +(-1.98224 - 11.5533i) 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} +(4.14418 - 3.13461i) q^{54} +2.27752 q^{55} -2.54025 q^{56} +(0.720857 - 2.26801i) q^{57} +(-4.18840 - 4.18840i) q^{58} +(-5.99556 - 5.99556i) q^{59} +(-2.05000 + 6.44983i) q^{60} +9.34533 q^{61} -9.19215 q^{62} +(-1.28868 - 7.51099i) 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} +(-2.95680 + 0.507306i) 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} +(-1.33273 + 4.19313i) 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} +(-4.82264 + 15.1733i) q^{93} +7.79393 q^{94} +5.36867 q^{95} +(1.65068 + 0.524648i) q^{96} +(0.433704 + 0.433704i) q^{97} +(-0.386893 - 0.386893i) q^{98} +(1.72345 - 0.295697i) 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 \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

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

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 −0.273849 + 0.961773i \(0.588297\pi\)
−0.961773 + 0.273849i \(0.911703\pi\)
\(6\) −1.65068 − 0.524648i −0.673887 − 0.214186i
\(7\) −1.79623 + 1.79623i −0.678909 + 0.678909i −0.959753 − 0.280844i \(-0.909386\pi\)
0.280844 + 0.959753i \(0.409386\pi\)
\(8\) 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 \(-0.278007\pi\)
−0.766506 + 0.642237i \(0.778007\pi\)
\(12\) 1.53819 − 0.796225i 0.444037 − 0.229850i
\(13\) 0 0
\(14\) − 2.54025i − 0.678909i
\(15\) −6.44983 − 2.05000i −1.66534 − 0.529307i
\(16\) −1.00000 −0.250000
\(17\) −1.09400 −0.265334 −0.132667 − 0.991161i \(-0.542354\pi\)
−0.132667 + 0.991161i \(0.542354\pi\)
\(18\) −0.507306 − 2.95680i −0.119573 − 0.696923i
\(19\) −0.971553 − 0.971553i −0.222890 − 0.222890i 0.586825 − 0.809714i \(-0.300378\pi\)
−0.809714 + 0.586825i \(0.800378\pi\)
\(20\) 2.76293 + 2.76293i 0.617811 + 0.617811i
\(21\) −4.19313 − 1.33273i −0.915017 − 0.290826i
\(22\) 0.582877 0.124270
\(23\) 1.75292 0.365509 0.182755 − 0.983159i \(-0.441499\pi\)
0.182755 + 0.983159i \(0.441499\pi\)
\(24\) −0.524648 + 1.65068i −0.107093 + 0.336944i
\(25\) − 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 0.982816 + 0.184588i \(0.0590950\pi\)
0.184588 + 0.982816i \(0.440905\pi\)
\(32\) 0.707107 − 0.707107i 0.125000 − 0.125000i
\(33\) 0.305805 − 0.962144i 0.0532339 − 0.167488i
\(34\) 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 −0.503957 − 0.863729i \(-0.668123\pi\)
0.863729 + 0.503957i \(0.168123\pi\)
\(38\) 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 \(-0.614225\pi\)
0.936302 + 0.351197i \(0.114225\pi\)
\(42\) 3.90738 − 2.02261i 0.602922 − 0.312095i
\(43\) − 3.76778i − 0.574581i −0.957844 − 0.287290i \(-0.907246\pi\)
0.957844 − 0.287290i \(-0.0927545\pi\)
\(44\) −0.412157 + 0.412157i −0.0621349 + 0.0621349i
\(45\) −1.98224 − 11.5533i −0.295494 − 1.72227i
\(46\) −1.23950 + 1.23950i −0.182755 + 0.182755i
\(47\) −5.51114 − 5.51114i −0.803883 − 0.803883i 0.179817 − 0.983700i \(-0.442449\pi\)
−0.983700 + 0.179817i \(0.942449\pi\)
\(48\) −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 \(0.932951\pi\)
\(54\) 4.14418 − 3.13461i 0.563952 − 0.426566i
\(55\) 2.27752 0.307101
\(56\) −2.54025 −0.339455
\(57\) 0.720857 − 2.26801i 0.0954799 − 0.300405i
\(58\) −4.18840 − 4.18840i −0.549965 − 0.549965i
\(59\) −5.99556 − 5.99556i −0.780556 − 0.780556i 0.199369 − 0.979925i \(-0.436111\pi\)
−0.979925 + 0.199369i \(0.936111\pi\)
\(60\) −2.05000 + 6.44983i −0.264653 + 0.832670i
\(61\) 9.34533 1.19655 0.598273 − 0.801292i \(-0.295854\pi\)
0.598273 + 0.801292i \(0.295854\pi\)
\(62\) −9.19215 −1.16740
\(63\) −1.28868 − 7.51099i −0.162359 − 0.946296i
\(64\) 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 0.932204 − 0.361932i \(-0.117883\pi\)
−0.361932 + 0.932204i \(0.617883\pi\)
\(68\) 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 \(-0.733929\pi\)
0.741891 + 0.670520i \(0.233929\pi\)
\(72\) −2.95680 + 0.507306i −0.348462 + 0.0597866i
\(73\) −5.18078 + 5.18078i −0.606365 + 0.606365i −0.941994 − 0.335629i \(-0.891051\pi\)
0.335629 + 0.941994i \(0.391051\pi\)
\(74\) 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 \(-0.880998\pi\)
0.365208 + 0.930926i \(0.380998\pi\)
\(84\) −1.33273 + 4.19313i −0.145413 + 0.457508i
\(85\) 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 \(-0.421678\pi\)
−0.969881 + 0.243580i \(0.921678\pi\)
\(90\) 9.57108 + 6.76778i 1.00888 + 0.713386i
\(91\) 0 0
\(92\) − 1.75292i − 0.182755i
\(93\) −4.82264 + 15.1733i −0.500084 + 1.57340i
\(94\) 7.79393 0.803883
\(95\) 5.36867 0.550814
\(96\) 1.65068 + 0.524648i 0.168472 + 0.0535466i
\(97\) 0.433704 + 0.433704i 0.0440360 + 0.0440360i 0.728782 − 0.684746i \(-0.240087\pi\)
−0.684746 + 0.728782i \(0.740087\pi\)
\(98\) −0.386893 − 0.386893i −0.0390821 − 0.0390821i
\(99\) 1.72345 − 0.295697i 0.173213 − 0.0297187i
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.437.3 16
3.2 odd 2 inner 1014.2.g.c.437.7 16
13.2 odd 12 78.2.k.a.71.1 yes 16
13.5 odd 4 inner 1014.2.g.c.239.7 16
13.8 odd 4 1014.2.g.d.239.3 16
13.9 even 3 78.2.k.a.11.4 yes 16
13.12 even 2 1014.2.g.d.437.7 16
39.2 even 12 78.2.k.a.71.4 yes 16
39.5 even 4 inner 1014.2.g.c.239.3 16
39.8 even 4 1014.2.g.d.239.7 16
39.35 odd 6 78.2.k.a.11.1 ✓ 16
39.38 odd 2 1014.2.g.d.437.3 16
52.15 even 12 624.2.cn.d.305.4 16
52.35 odd 6 624.2.cn.d.401.2 16
156.35 even 6 624.2.cn.d.401.4 16
156.119 odd 12 624.2.cn.d.305.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.k.a.11.1 ✓ 16 39.35 odd 6
78.2.k.a.11.4 yes 16 13.9 even 3
78.2.k.a.71.1 yes 16 13.2 odd 12
78.2.k.a.71.4 yes 16 39.2 even 12
624.2.cn.d.305.2 16 156.119 odd 12
624.2.cn.d.305.4 16 52.15 even 12
624.2.cn.d.401.2 16 52.35 odd 6
624.2.cn.d.401.4 16 156.35 even 6
1014.2.g.c.239.3 16 39.5 even 4 inner
1014.2.g.c.239.7 16 13.5 odd 4 inner
1014.2.g.c.437.3 16 1.1 even 1 trivial
1014.2.g.c.437.7 16 3.2 odd 2 inner
1014.2.g.d.239.3 16 13.8 odd 4
1014.2.g.d.239.7 16 39.8 even 4
1014.2.g.d.437.3 16 39.38 odd 2
1014.2.g.d.437.7 16 13.12 even 2