Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [58,2,Mod(5,58)] 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("58.5"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(58, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([11])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 58.e (of order \(14\), degree \(6\), 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: \(0.463132331723\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{14})\)
Coefficient field: \(\Q(\zeta_{28})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label 9.1
Root \(-0.974928 + 0.222521i\) of defining polynomial
Character \(\chi\) \(=\) 58.9
Dual form 58.2.e.a.13.1

$q$-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+(-0.433884 - 0.900969i) q^{2} +(2.19064 - 1.74698i) q^{3} +(-0.623490 + 0.781831i) q^{4} +(-3.43956 + 1.65640i) q^{5} +(-2.52446 - 1.21572i) q^{6} +(1.36428 + 1.71075i) q^{7} +(0.974928 + 0.222521i) q^{8} +(1.07942 - 4.72923i) q^{9} +(2.98474 + 2.38025i) q^{10} +(-0.0647379 + 0.0147760i) q^{11} +2.80194i q^{12} +(-0.157018 - 0.687941i) q^{13} +(0.949394 - 1.97144i) q^{14} +(-4.64114 + 9.63743i) q^{15} +(-0.222521 - 0.974928i) q^{16} +3.46337i q^{17} +(-4.72923 + 1.07942i) q^{18} +(-2.15993 - 1.72248i) q^{19} +(0.849501 - 3.72191i) q^{20} +(5.97729 + 1.36428i) q^{21} +(0.0414015 + 0.0519158i) q^{22} +(-5.68374 - 2.73715i) q^{23} +(2.52446 - 1.21572i) q^{24} +(5.96945 - 7.48545i) q^{25} +(-0.551686 + 0.439955i) q^{26} +(-2.25011 - 4.67241i) q^{27} -2.18813 q^{28} +(2.16940 - 4.92886i) q^{29} +10.6967 q^{30} +(3.24823 + 6.74502i) q^{31} +(-0.781831 + 0.623490i) q^{32} +(-0.116004 + 0.145465i) q^{33} +(3.12039 - 1.50270i) q^{34} +(-7.52621 - 3.62443i) q^{35} +(3.02446 + 3.79255i) q^{36} +(-8.31592 - 1.89805i) q^{37} +(-0.614747 + 2.69338i) q^{38} +(-1.54579 - 1.23273i) q^{39} +(-3.72191 + 0.849501i) q^{40} -2.48403i q^{41} +(-1.36428 - 5.97729i) q^{42} +(-0.624636 + 1.29707i) q^{43} +(0.0288111 - 0.0598268i) q^{44} +(4.12081 + 18.0544i) q^{45} +6.30848i q^{46} +(8.77125 - 2.00198i) q^{47} +(-2.19064 - 1.74698i) q^{48} +(0.492235 - 2.15662i) q^{49} +(-9.33420 - 2.13047i) q^{50} +(6.05044 + 7.58702i) q^{51} +(0.635753 + 0.306163i) q^{52} +(2.90835 - 1.40059i) q^{53} +(-3.23341 + 4.05456i) q^{54} +(0.198195 - 0.158055i) q^{55} +(0.949394 + 1.97144i) q^{56} -7.74077 q^{57} +(-5.38202 + 0.183989i) q^{58} +1.24060 q^{59} +(-4.64114 - 9.63743i) q^{60} +(2.61945 - 2.08894i) q^{61} +(4.66770 - 5.85311i) q^{62} +(9.56316 - 4.60537i) q^{63} +(0.900969 + 0.433884i) q^{64} +(1.67958 + 2.10613i) q^{65} +(0.181392 + 0.0414015i) q^{66} +(-1.56082 + 6.83840i) q^{67} +(-2.70778 - 2.15938i) q^{68} +(-17.2328 + 3.93327i) q^{69} +8.35346i q^{70} +(2.24599 + 9.84033i) q^{71} +(2.10471 - 4.37047i) q^{72} +(1.31885 - 2.73862i) q^{73} +(1.89805 + 8.31592i) q^{74} -26.8264i q^{75} +(2.69338 - 0.614747i) q^{76} +(-0.113599 - 0.0905918i) q^{77} +(-0.439955 + 1.92757i) q^{78} +(2.17779 + 0.497066i) q^{79} +(2.38025 + 2.98474i) q^{80} +(0.0196143 + 0.00944576i) q^{81} +(-2.23803 + 1.07778i) q^{82} +(-3.82548 + 4.79700i) q^{83} +(-4.79341 + 3.82262i) q^{84} +(-5.73675 - 11.9125i) q^{85} +1.43964 q^{86} +(-3.85824 - 14.5873i) q^{87} -0.0664028 q^{88} +(7.59564 + 15.7725i) q^{89} +(14.4785 - 11.5462i) q^{90} +(0.962679 - 1.20716i) q^{91} +(5.68374 - 2.73715i) q^{92} +(18.8991 + 9.10134i) q^{93} +(-5.60942 - 7.03400i) q^{94} +(10.2823 + 2.34687i) q^{95} +(-0.623490 + 2.73169i) q^{96} +(-1.72181 - 1.37309i) q^{97} +(-2.15662 + 0.492235i) q^{98} +0.322110i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{4} - 2 q^{5} - 12 q^{6} + 4 q^{7} - 4 q^{9} - 26 q^{13} - 14 q^{15} - 2 q^{16} + 2 q^{20} + 14 q^{21} + 4 q^{22} - 16 q^{23} + 12 q^{24} + 22 q^{25} + 14 q^{26} - 4 q^{28} + 18 q^{29} + 16 q^{30}+ \cdots - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).

\(n\) \(31\)
\(\chi(n)\) \(e\left(\frac{5}{14}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.433884 − 0.900969i −0.306802 − 0.637081i
\(3\) 2.19064 − 1.74698i 1.26477 − 1.00862i 0.265763 − 0.964039i \(-0.414376\pi\)
0.999006 − 0.0445806i \(-0.0141952\pi\)
\(4\) −0.623490 + 0.781831i −0.311745 + 0.390916i
\(5\) −3.43956 + 1.65640i −1.53822 + 0.740767i −0.995098 − 0.0988888i \(-0.968471\pi\)
−0.543119 + 0.839655i \(0.682757\pi\)
\(6\) −2.52446 − 1.21572i −1.03061 − 0.496314i
\(7\) 1.36428 + 1.71075i 0.515648 + 0.646602i 0.969679 − 0.244384i \(-0.0785857\pi\)
−0.454030 + 0.890986i \(0.650014\pi\)
\(8\) 0.974928 + 0.222521i 0.344689 + 0.0786730i
\(9\) 1.07942 − 4.72923i 0.359806 − 1.57641i
\(10\) 2.98474 + 2.38025i 0.943857 + 0.752701i
\(11\) −0.0647379 + 0.0147760i −0.0195192 + 0.00445513i −0.232269 − 0.972651i \(-0.574615\pi\)
0.212750 + 0.977107i \(0.431758\pi\)
\(12\) 2.80194i 0.808850i
\(13\) −0.157018 − 0.687941i −0.0435490 − 0.190801i 0.948475 − 0.316853i \(-0.102626\pi\)
−0.992024 + 0.126053i \(0.959769\pi\)
\(14\) 0.949394 − 1.97144i 0.253736 − 0.526889i
\(15\) −4.64114 + 9.63743i −1.19834 + 2.48837i
\(16\) −0.222521 − 0.974928i −0.0556302 − 0.243732i
\(17\) 3.46337i 0.839992i 0.907526 + 0.419996i \(0.137968\pi\)
−0.907526 + 0.419996i \(0.862032\pi\)
\(18\) −4.72923 + 1.07942i −1.11469 + 0.254421i
\(19\) −2.15993 − 1.72248i −0.495521 − 0.395165i 0.343595 − 0.939118i \(-0.388355\pi\)
−0.839115 + 0.543953i \(0.816927\pi\)
\(20\) 0.849501 − 3.72191i 0.189954 − 0.832244i
\(21\) 5.97729 + 1.36428i 1.30435 + 0.297710i
\(22\) 0.0414015 + 0.0519158i 0.00882682 + 0.0110685i
\(23\) −5.68374 − 2.73715i −1.18514 − 0.570734i −0.265736 − 0.964046i \(-0.585615\pi\)
−0.919405 + 0.393312i \(0.871329\pi\)
\(24\) 2.52446 − 1.21572i 0.515303 − 0.248157i
\(25\) 5.96945 − 7.48545i 1.19389 − 1.49709i
\(26\) −0.551686 + 0.439955i −0.108195 + 0.0862823i
\(27\) −2.25011 − 4.67241i −0.433034 − 0.899205i
\(28\) −2.18813 −0.413518
\(29\) 2.16940 − 4.92886i 0.402848 − 0.915267i
\(30\) 10.6967 1.95295
\(31\) 3.24823 + 6.74502i 0.583399 + 1.21144i 0.958669 + 0.284525i \(0.0918359\pi\)
−0.375269 + 0.926916i \(0.622450\pi\)
\(32\) −0.781831 + 0.623490i −0.138210 + 0.110218i
\(33\) −0.116004 + 0.145465i −0.0201938 + 0.0253222i
\(34\) 3.12039 − 1.50270i 0.535143 − 0.257711i
\(35\) −7.52621 − 3.62443i −1.27216 − 0.612640i
\(36\) 3.02446 + 3.79255i 0.504076 + 0.632092i
\(37\) −8.31592 − 1.89805i −1.36713 − 0.312038i −0.524902 − 0.851162i \(-0.675898\pi\)
−0.842226 + 0.539124i \(0.818755\pi\)
\(38\) −0.614747 + 2.69338i −0.0997251 + 0.436924i
\(39\) −1.54579 − 1.23273i −0.247524 − 0.197394i
\(40\) −3.72191 + 0.849501i −0.588485 + 0.134318i
\(41\) − 2.48403i − 0.387940i −0.981007 − 0.193970i \(-0.937864\pi\)
0.981007 − 0.193970i \(-0.0621364\pi\)
\(42\) −1.36428 − 5.97729i −0.210513 − 0.922316i
\(43\) −0.624636 + 1.29707i −0.0952560 + 0.197801i −0.943152 − 0.332361i \(-0.892155\pi\)
0.847896 + 0.530162i \(0.177869\pi\)
\(44\) 0.0288111 − 0.0598268i 0.00434343 − 0.00901924i
\(45\) 4.12081 + 18.0544i 0.614294 + 2.69140i
\(46\) 6.30848i 0.930134i
\(47\) 8.77125 − 2.00198i 1.27942 − 0.292019i 0.471790 − 0.881711i \(-0.343608\pi\)
0.807628 + 0.589692i \(0.200751\pi\)
\(48\) −2.19064 − 1.74698i −0.316192 − 0.252155i
\(49\) 0.492235 − 2.15662i 0.0703193 − 0.308089i
\(50\) −9.33420 − 2.13047i −1.32006 − 0.301294i
\(51\) 6.05044 + 7.58702i 0.847232 + 1.06239i
\(52\) 0.635753 + 0.306163i 0.0881631 + 0.0424571i
\(53\) 2.90835 − 1.40059i 0.399493 − 0.192386i −0.223340 − 0.974741i \(-0.571696\pi\)
0.622833 + 0.782355i \(0.285982\pi\)
\(54\) −3.23341 + 4.05456i −0.440011 + 0.551756i
\(55\) 0.198195 − 0.158055i 0.0267246 − 0.0213122i
\(56\) 0.949394 + 1.97144i 0.126868 + 0.263444i
\(57\) −7.74077 −1.02529
\(58\) −5.38202 + 0.183989i −0.706694 + 0.0241590i
\(59\) 1.24060 0.161513 0.0807564 − 0.996734i \(-0.474266\pi\)
0.0807564 + 0.996734i \(0.474266\pi\)
\(60\) −4.64114 − 9.63743i −0.599169 − 1.24419i
\(61\) 2.61945 − 2.08894i 0.335386 − 0.267462i −0.441286 − 0.897366i \(-0.645478\pi\)
0.776672 + 0.629905i \(0.216906\pi\)
\(62\) 4.66770 − 5.85311i 0.592798 − 0.743345i
\(63\) 9.56316 − 4.60537i 1.20484 − 0.580223i
\(64\) 0.900969 + 0.433884i 0.112621 + 0.0542355i
\(65\) 1.67958 + 2.10613i 0.208327 + 0.261233i
\(66\) 0.181392 + 0.0414015i 0.0223278 + 0.00509617i
\(67\) −1.56082 + 6.83840i −0.190685 + 0.835444i 0.785562 + 0.618783i \(0.212374\pi\)
−0.976247 + 0.216661i \(0.930483\pi\)
\(68\) −2.70778 − 2.15938i −0.328366 − 0.261863i
\(69\) −17.2328 + 3.93327i −2.07458 + 0.473510i
\(70\) 8.35346i 0.998429i
\(71\) 2.24599 + 9.84033i 0.266550 + 1.16783i 0.913997 + 0.405721i \(0.132980\pi\)
−0.647447 + 0.762111i \(0.724163\pi\)
\(72\) 2.10471 − 4.37047i 0.248042 − 0.515065i
\(73\) 1.31885 − 2.73862i 0.154360 − 0.320531i −0.809420 − 0.587230i \(-0.800219\pi\)
0.963780 + 0.266698i \(0.0859328\pi\)
\(74\) 1.89805 + 8.31592i 0.220644 + 0.966706i
\(75\) − 26.8264i − 3.09765i
\(76\) 2.69338 − 0.614747i 0.308952 − 0.0705163i
\(77\) −0.113599 − 0.0905918i −0.0129458 − 0.0103239i
\(78\) −0.439955 + 1.92757i −0.0498151 + 0.218254i
\(79\) 2.17779 + 0.497066i 0.245020 + 0.0559243i 0.343267 − 0.939238i \(-0.388466\pi\)
−0.0982469 + 0.995162i \(0.531323\pi\)
\(80\) 2.38025 + 2.98474i 0.266120 + 0.333704i
\(81\) 0.0196143 + 0.00944576i 0.00217937 + 0.00104953i
\(82\) −2.23803 + 1.07778i −0.247149 + 0.119021i
\(83\) −3.82548 + 4.79700i −0.419901 + 0.526539i −0.946123 − 0.323808i \(-0.895037\pi\)
0.526222 + 0.850347i \(0.323608\pi\)
\(84\) −4.79341 + 3.82262i −0.523004 + 0.417082i
\(85\) −5.73675 − 11.9125i −0.622238 − 1.29209i
\(86\) 1.43964 0.155240
\(87\) −3.85824 − 14.5873i −0.413646 − 1.56392i
\(88\) −0.0664028 −0.00707856
\(89\) 7.59564 + 15.7725i 0.805136 + 1.67188i 0.738645 + 0.674095i \(0.235466\pi\)
0.0664914 + 0.997787i \(0.478820\pi\)
\(90\) 14.4785 − 11.5462i 1.52617 − 1.21708i
\(91\) 0.962679 − 1.20716i 0.100916 − 0.126545i
\(92\) 5.68374 − 2.73715i 0.592571 − 0.285367i
\(93\) 18.8991 + 9.10134i 1.95975 + 0.943765i
\(94\) −5.60942 − 7.03400i −0.578568 − 0.725501i
\(95\) 10.2823 + 2.34687i 1.05494 + 0.240784i
\(96\) −0.623490 + 2.73169i −0.0636347 + 0.278802i
\(97\) −1.72181 − 1.37309i −0.174823 − 0.139417i 0.532169 − 0.846638i \(-0.321377\pi\)
−0.706992 + 0.707221i \(0.749948\pi\)
\(98\) −2.15662 + 0.492235i −0.217852 + 0.0497233i
\(99\) 0.322110i 0.0323733i
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 58.2.e.a.9.1 ✓ 12
3.2 odd 2 522.2.n.a.415.2 12
4.3 odd 2 464.2.y.c.241.1 12
29.4 even 14 1682.2.b.j.1681.6 12
29.10 odd 28 1682.2.a.r.1.5 6
29.13 even 14 inner 58.2.e.a.13.1 yes 12
29.19 odd 28 1682.2.a.s.1.1 6
29.25 even 7 1682.2.b.j.1681.8 12
87.71 odd 14 522.2.n.a.361.2 12
116.71 odd 14 464.2.y.c.129.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.e.a.9.1 ✓ 12 1.1 even 1 trivial
58.2.e.a.13.1 yes 12 29.13 even 14 inner
464.2.y.c.129.1 12 116.71 odd 14
464.2.y.c.241.1 12 4.3 odd 2
522.2.n.a.361.2 12 87.71 odd 14
522.2.n.a.415.2 12 3.2 odd 2
1682.2.a.r.1.5 6 29.10 odd 28
1682.2.a.s.1.1 6 29.19 odd 28
1682.2.b.j.1681.6 12 29.4 even 14
1682.2.b.j.1681.8 12 29.25 even 7