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(7,58)] 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("58.7"); 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([6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 58.d (of order \(7\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == 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 \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 13 x^{10} - 9 x^{9} - 5 x^{8} + 35 x^{7} + 197 x^{6} - 140 x^{5} - 80 x^{4} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label 45.1
Root \(1.52179 + 1.90827i\) of defining polynomial
Character \(\chi\) \(=\) 58.45
Dual form 58.2.d.b.49.1

$q$-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+(-0.900969 - 0.433884i) q^{2} +(-1.52179 + 1.90827i) q^{3} +(0.623490 + 0.781831i) q^{4} +(2.60002 + 1.25211i) q^{5} +(2.19905 - 1.05901i) q^{6} +(-1.89765 + 2.37957i) q^{7} +(-0.222521 - 0.974928i) q^{8} +(-0.658071 - 2.88320i) q^{9} +(-1.79927 - 2.25622i) q^{10} +(1.22038 - 5.34685i) q^{11} -2.44077 q^{12} +(0.0239308 - 0.104847i) q^{13} +(2.74218 - 1.32056i) q^{14} +(-6.34605 + 3.05610i) q^{15} +(-0.222521 + 0.974928i) q^{16} +0.816005 q^{17} +(-0.658071 + 2.88320i) q^{18} +(1.27358 + 1.59701i) q^{19} +(0.642153 + 2.81346i) q^{20} +(-1.65304 - 7.24243i) q^{21} +(-3.41944 + 4.28784i) q^{22} +(8.25746 - 3.97659i) q^{23} +(2.19905 + 1.05901i) q^{24} +(2.07491 + 2.60185i) q^{25} +(-0.0670525 + 0.0840812i) q^{26} +(-0.0938049 - 0.0451741i) q^{27} -3.04359 q^{28} +(-5.37657 + 0.304047i) q^{29} +7.04359 q^{30} +(3.10086 + 1.49330i) q^{31} +(0.623490 - 0.781831i) q^{32} +(8.34605 + 10.4656i) q^{33} +(-0.735195 - 0.354051i) q^{34} +(-7.91340 + 3.81089i) q^{35} +(1.84387 - 2.31214i) q^{36} +(-1.31456 - 5.75946i) q^{37} +(-0.454534 - 1.99144i) q^{38} +(0.163659 + 0.205223i) q^{39} +(0.642153 - 2.81346i) q^{40} -2.43376 q^{41} +(-1.65304 + 7.24243i) q^{42} +(-4.94123 + 2.37957i) q^{43} +(4.94123 - 2.37957i) q^{44} +(1.89907 - 8.32035i) q^{45} -9.16509 q^{46} +(-1.31510 + 5.76182i) q^{47} +(-1.52179 - 1.90827i) q^{48} +(-0.503658 - 2.20667i) q^{49} +(-0.740526 - 3.24446i) q^{50} +(-1.24179 + 1.55716i) q^{51} +(0.0968937 - 0.0466615i) q^{52} +(-3.55945 - 1.71414i) q^{53} +(0.0649150 + 0.0814008i) q^{54} +(9.86785 - 12.3739i) q^{55} +(2.74218 + 1.32056i) q^{56} -4.98565 q^{57} +(4.97605 + 2.05887i) q^{58} +1.13359 q^{59} +(-6.34605 - 3.05610i) q^{60} +(5.09132 - 6.38431i) q^{61} +(-2.14586 - 2.69083i) q^{62} +(8.10956 + 3.90536i) q^{63} +(-0.900969 + 0.433884i) q^{64} +(0.193501 - 0.242642i) q^{65} +(-2.97867 - 13.0504i) q^{66} +(-0.212822 - 0.932434i) q^{67} +(0.508771 + 0.637978i) q^{68} +(-4.97776 + 21.8090i) q^{69} +8.78321 q^{70} +(-0.531778 + 2.32987i) q^{71} +(-2.66447 + 1.28314i) q^{72} +(-8.01331 + 3.85900i) q^{73} +(-1.31456 + 5.75946i) q^{74} -8.12261 q^{75} +(-0.454534 + 1.99144i) q^{76} +(10.4074 + 13.0504i) q^{77} +(-0.0584094 - 0.255908i) q^{78} +(0.934726 + 4.09530i) q^{79} +(-1.79927 + 2.25622i) q^{80} +(8.22238 - 3.95969i) q^{81} +(2.19274 + 1.05597i) q^{82} +(-9.64738 - 12.0974i) q^{83} +(4.63171 - 5.80798i) q^{84} +(2.12163 + 1.02172i) q^{85} +5.48435 q^{86} +(7.60183 - 10.7226i) q^{87} -5.48435 q^{88} +(-4.99275 - 2.40438i) q^{89} +(-5.32107 + 6.67241i) q^{90} +(0.204080 + 0.255908i) q^{91} +(8.25746 + 3.97659i) q^{92} +(-7.56848 + 3.64479i) q^{93} +(3.68482 - 4.62062i) q^{94} +(1.31170 + 5.74692i) q^{95} +(0.543122 + 2.37957i) q^{96} +(7.43305 + 9.32075i) q^{97} +(-0.503658 + 2.20667i) q^{98} -16.2191 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} - 3 q^{3} - 2 q^{4} + 4 q^{6} + q^{7} - 2 q^{8} - 11 q^{9} - 7 q^{10} - 2 q^{11} + 4 q^{12} + q^{13} + q^{14} - 9 q^{15} - 2 q^{16} - 12 q^{17} - 11 q^{18} - 6 q^{19} + 7 q^{20} - 13 q^{21}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(31\)
\(\chi(n)\) \(e\left(\frac{1}{7}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.900969 − 0.433884i −0.637081 − 0.306802i
\(3\) −1.52179 + 1.90827i −0.878608 + 1.10174i 0.115496 + 0.993308i \(0.463154\pi\)
−0.994104 + 0.108431i \(0.965417\pi\)
\(4\) 0.623490 + 0.781831i 0.311745 + 0.390916i
\(5\) 2.60002 + 1.25211i 1.16277 + 0.559959i 0.912844 − 0.408307i \(-0.133881\pi\)
0.249922 + 0.968266i \(0.419595\pi\)
\(6\) 2.19905 − 1.05901i 0.897760 − 0.432339i
\(7\) −1.89765 + 2.37957i −0.717242 + 0.899394i −0.998178 − 0.0603330i \(-0.980784\pi\)
0.280936 + 0.959727i \(0.409355\pi\)
\(8\) −0.222521 − 0.974928i −0.0786730 − 0.344689i
\(9\) −0.658071 − 2.88320i −0.219357 − 0.961065i
\(10\) −1.79927 − 2.25622i −0.568980 − 0.713478i
\(11\) 1.22038 − 5.34685i 0.367959 − 1.61214i −0.364417 − 0.931236i \(-0.618732\pi\)
0.732377 − 0.680900i \(-0.238411\pi\)
\(12\) −2.44077 −0.704589
\(13\) 0.0239308 − 0.104847i 0.00663720 − 0.0290795i −0.971501 − 0.237036i \(-0.923824\pi\)
0.978138 + 0.207956i \(0.0666812\pi\)
\(14\) 2.74218 − 1.32056i 0.732878 − 0.352935i
\(15\) −6.34605 + 3.05610i −1.63854 + 0.789081i
\(16\) −0.222521 + 0.974928i −0.0556302 + 0.243732i
\(17\) 0.816005 0.197910 0.0989551 − 0.995092i \(-0.468450\pi\)
0.0989551 + 0.995092i \(0.468450\pi\)
\(18\) −0.658071 + 2.88320i −0.155109 + 0.679576i
\(19\) 1.27358 + 1.59701i 0.292178 + 0.366380i 0.906156 − 0.422943i \(-0.139003\pi\)
−0.613978 + 0.789323i \(0.710432\pi\)
\(20\) 0.642153 + 2.81346i 0.143590 + 0.629108i
\(21\) −1.65304 − 7.24243i −0.360722 − 1.58043i
\(22\) −3.41944 + 4.28784i −0.729027 + 0.914171i
\(23\) 8.25746 − 3.97659i 1.72180 − 0.829175i 0.732946 − 0.680286i \(-0.238145\pi\)
0.988854 − 0.148889i \(-0.0475697\pi\)
\(24\) 2.19905 + 1.05901i 0.448880 + 0.216169i
\(25\) 2.07491 + 2.60185i 0.414981 + 0.520370i
\(26\) −0.0670525 + 0.0840812i −0.0131501 + 0.0164897i
\(27\) −0.0938049 − 0.0451741i −0.0180528 − 0.00869375i
\(28\) −3.04359 −0.575184
\(29\) −5.37657 + 0.304047i −0.998405 + 0.0564602i
\(30\) 7.04359 1.28598
\(31\) 3.10086 + 1.49330i 0.556931 + 0.268204i 0.691106 − 0.722754i \(-0.257124\pi\)
−0.134174 + 0.990958i \(0.542838\pi\)
\(32\) 0.623490 − 0.781831i 0.110218 − 0.138210i
\(33\) 8.34605 + 10.4656i 1.45286 + 1.82183i
\(34\) −0.735195 − 0.354051i −0.126085 − 0.0607193i
\(35\) −7.91340 + 3.81089i −1.33761 + 0.644158i
\(36\) 1.84387 − 2.31214i 0.307312 − 0.385357i
\(37\) −1.31456 − 5.75946i −0.216112 − 0.946850i −0.960320 − 0.278901i \(-0.910030\pi\)
0.744207 − 0.667949i \(-0.232827\pi\)
\(38\) −0.454534 − 1.99144i −0.0737351 − 0.323055i
\(39\) 0.163659 + 0.205223i 0.0262065 + 0.0328619i
\(40\) 0.642153 − 2.81346i 0.101533 − 0.444846i
\(41\) −2.43376 −0.380090 −0.190045 − 0.981775i \(-0.560863\pi\)
−0.190045 + 0.981775i \(0.560863\pi\)
\(42\) −1.65304 + 7.24243i −0.255069 + 1.11753i
\(43\) −4.94123 + 2.37957i −0.753531 + 0.362881i −0.770890 − 0.636968i \(-0.780188\pi\)
0.0173595 + 0.999849i \(0.494474\pi\)
\(44\) 4.94123 − 2.37957i 0.744919 − 0.358734i
\(45\) 1.89907 − 8.32035i 0.283096 − 1.24033i
\(46\) −9.16509 −1.35132
\(47\) −1.31510 + 5.76182i −0.191827 + 0.840448i 0.783800 + 0.621013i \(0.213279\pi\)
−0.975627 + 0.219435i \(0.929579\pi\)
\(48\) −1.52179 − 1.90827i −0.219652 − 0.275435i
\(49\) −0.503658 − 2.20667i −0.0719512 − 0.315239i
\(50\) −0.740526 − 3.24446i −0.104726 − 0.458835i
\(51\) −1.24179 + 1.55716i −0.173885 + 0.218045i
\(52\) 0.0968937 − 0.0466615i 0.0134367 − 0.00647079i
\(53\) −3.55945 − 1.71414i −0.488929 − 0.235456i 0.173142 − 0.984897i \(-0.444608\pi\)
−0.662071 + 0.749441i \(0.730322\pi\)
\(54\) 0.0649150 + 0.0814008i 0.00883381 + 0.0110773i
\(55\) 9.86785 − 12.3739i 1.33058 − 1.66849i
\(56\) 2.74218 + 1.32056i 0.366439 + 0.176468i
\(57\) −4.98565 −0.660365
\(58\) 4.97605 + 2.05887i 0.653387 + 0.270343i
\(59\) 1.13359 0.147581 0.0737904 − 0.997274i \(-0.476490\pi\)
0.0737904 + 0.997274i \(0.476490\pi\)
\(60\) −6.34605 − 3.05610i −0.819272 − 0.394541i
\(61\) 5.09132 − 6.38431i 0.651876 − 0.817427i −0.340555 − 0.940224i \(-0.610615\pi\)
0.992432 + 0.122797i \(0.0391865\pi\)
\(62\) −2.14586 − 2.69083i −0.272525 − 0.341735i
\(63\) 8.10956 + 3.90536i 1.02171 + 0.492029i
\(64\) −0.900969 + 0.433884i −0.112621 + 0.0542355i
\(65\) 0.193501 − 0.242642i 0.0240008 − 0.0300961i
\(66\) −2.97867 − 13.0504i −0.366649 − 1.60639i
\(67\) −0.212822 − 0.932434i −0.0260003 − 0.113915i 0.960263 − 0.279098i \(-0.0900355\pi\)
−0.986263 + 0.165183i \(0.947178\pi\)
\(68\) 0.508771 + 0.637978i 0.0616975 + 0.0773662i
\(69\) −4.97776 + 21.8090i −0.599252 + 2.62549i
\(70\) 8.78321 1.04979
\(71\) −0.531778 + 2.32987i −0.0631104 + 0.276505i −0.996631 − 0.0820198i \(-0.973863\pi\)
0.933520 + 0.358525i \(0.116720\pi\)
\(72\) −2.66447 + 1.28314i −0.314011 + 0.151220i
\(73\) −8.01331 + 3.85900i −0.937886 + 0.451662i −0.839423 − 0.543478i \(-0.817107\pi\)
−0.0984633 + 0.995141i \(0.531393\pi\)
\(74\) −1.31456 + 5.75946i −0.152814 + 0.669524i
\(75\) −8.12261 −0.937918
\(76\) −0.454534 + 1.99144i −0.0521386 + 0.228434i
\(77\) 10.4074 + 13.0504i 1.18603 + 1.48723i
\(78\) −0.0584094 − 0.255908i −0.00661356 − 0.0289759i
\(79\) 0.934726 + 4.09530i 0.105165 + 0.460758i 0.999900 + 0.0141598i \(0.00450735\pi\)
−0.894735 + 0.446598i \(0.852636\pi\)
\(80\) −1.79927 + 2.25622i −0.201165 + 0.252253i
\(81\) 8.22238 − 3.95969i 0.913598 − 0.439965i
\(82\) 2.19274 + 1.05597i 0.242148 + 0.116612i
\(83\) −9.64738 − 12.0974i −1.05894 − 1.32787i −0.942329 − 0.334688i \(-0.891369\pi\)
−0.116609 − 0.993178i \(-0.537202\pi\)
\(84\) 4.63171 − 5.80798i 0.505361 − 0.633703i
\(85\) 2.12163 + 1.02172i 0.230123 + 0.110822i
\(86\) 5.48435 0.591393
\(87\) 7.60183 − 10.7226i 0.815002 − 1.14959i
\(88\) −5.48435 −0.584634
\(89\) −4.99275 − 2.40438i −0.529231 − 0.254864i 0.150133 − 0.988666i \(-0.452030\pi\)
−0.679364 + 0.733802i \(0.737744\pi\)
\(90\) −5.32107 + 6.67241i −0.560890 + 0.703333i
\(91\) 0.204080 + 0.255908i 0.0213934 + 0.0268265i
\(92\) 8.25746 + 3.97659i 0.860900 + 0.414588i
\(93\) −7.56848 + 3.64479i −0.784815 + 0.377947i
\(94\) 3.68482 − 4.62062i 0.380060 − 0.476581i
\(95\) 1.31170 + 5.74692i 0.134577 + 0.589622i
\(96\) 0.543122 + 2.37957i 0.0554321 + 0.242864i
\(97\) 7.43305 + 9.32075i 0.754712 + 0.946379i 0.999732 − 0.0231388i \(-0.00736597\pi\)
−0.245020 + 0.969518i \(0.578795\pi\)
\(98\) −0.503658 + 2.20667i −0.0508772 + 0.222907i
\(99\) −16.2191 −1.63008
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.d.b.45.1 ✓ 12
3.2 odd 2 522.2.k.h.451.1 12
4.3 odd 2 464.2.u.h.161.2 12
29.3 odd 28 1682.2.b.i.1681.6 12
29.7 even 7 1682.2.a.t.1.1 6
29.20 even 7 inner 58.2.d.b.49.1 yes 12
29.22 even 14 1682.2.a.q.1.6 6
29.26 odd 28 1682.2.b.i.1681.7 12
87.20 odd 14 522.2.k.h.397.1 12
116.107 odd 14 464.2.u.h.49.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.45.1 ✓ 12 1.1 even 1 trivial
58.2.d.b.49.1 yes 12 29.20 even 7 inner
464.2.u.h.49.2 12 116.107 odd 14
464.2.u.h.161.2 12 4.3 odd 2
522.2.k.h.397.1 12 87.20 odd 14
522.2.k.h.451.1 12 3.2 odd 2
1682.2.a.q.1.6 6 29.22 even 14
1682.2.a.t.1.1 6 29.7 even 7
1682.2.b.i.1681.6 12 29.3 odd 28
1682.2.b.i.1681.7 12 29.26 odd 28