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 7.2
Root \(-1.56920 + 0.755686i\) of defining polynomial
Character \(\chi\) \(=\) 58.7
Dual form 58.2.d.b.25.2

$q$-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+(-0.222521 - 0.974928i) q^{2} +(1.56920 + 0.755686i) q^{3} +(-0.900969 + 0.433884i) q^{4} +(0.110081 + 0.482295i) q^{5} +(0.387560 - 1.69801i) q^{6} +(-2.82760 - 1.36170i) q^{7} +(0.623490 + 0.781831i) q^{8} +(0.0208506 + 0.0261458i) q^{9} +(0.445708 - 0.214642i) q^{10} +(0.870839 - 1.09200i) q^{11} -1.74168 q^{12} +(-3.56353 + 4.46852i) q^{13} +(-0.698359 + 3.05971i) q^{14} +(-0.191725 + 0.840003i) q^{15} +(0.623490 - 0.781831i) q^{16} +5.31800 q^{17} +(0.0208506 - 0.0261458i) q^{18} +(-4.47471 + 2.15491i) q^{19} +(-0.308439 - 0.386771i) q^{20} +(-3.40804 - 4.27355i) q^{21} +(-1.25840 - 0.606013i) q^{22} +(0.181045 - 0.793210i) q^{23} +(0.387560 + 1.69801i) q^{24} +(4.28435 - 2.06324i) q^{25} +(5.14944 + 2.47984i) q^{26} +(-1.14972 - 5.03725i) q^{27} +3.13840 q^{28} +(5.38501 + 0.0414712i) q^{29} +0.861605 q^{30} +(-1.41596 - 6.20373i) q^{31} +(-0.900969 - 0.433884i) q^{32} +(2.19173 - 1.05548i) q^{33} +(-1.18337 - 5.18466i) q^{34} +(0.345477 - 1.51363i) q^{35} +(-0.0301299 - 0.0145098i) q^{36} +(5.56630 + 6.97992i) q^{37} +(3.09660 + 3.88301i) q^{38} +(-8.96867 + 4.31909i) q^{39} +(-0.308439 + 0.386771i) q^{40} -4.01226 q^{41} +(-3.40804 + 4.27355i) q^{42} +(0.310799 - 1.36170i) q^{43} +(-0.310799 + 1.36170i) q^{44} +(-0.0103147 + 0.0129343i) q^{45} -0.813609 q^{46} +(-6.42079 + 8.05142i) q^{47} +(1.56920 - 0.755686i) q^{48} +(1.77665 + 2.22785i) q^{49} +(-2.96486 - 3.71782i) q^{50} +(8.34499 + 4.01873i) q^{51} +(1.27181 - 5.57215i) q^{52} +(-0.944288 - 4.13720i) q^{53} +(-4.65512 + 2.24179i) q^{54} +(0.622528 + 0.299794i) q^{55} +(-0.698359 - 3.05971i) q^{56} -8.65014 q^{57} +(-1.15784 - 5.25922i) q^{58} +11.1598 q^{59} +(-0.191725 - 0.840003i) q^{60} +(-4.38454 - 2.11149i) q^{61} +(-5.73311 + 2.76092i) q^{62} +(-0.0233543 - 0.102322i) q^{63} +(-0.222521 + 0.974928i) q^{64} +(-2.54742 - 1.22677i) q^{65} +(-1.51672 - 1.90191i) q^{66} +(4.45072 + 5.58102i) q^{67} +(-4.79135 + 2.30739i) q^{68} +(0.883513 - 1.10789i) q^{69} -1.55256 q^{70} +(3.76423 - 4.72019i) q^{71} +(-0.00744148 + 0.0326033i) q^{72} +(-2.73238 + 11.9713i) q^{73} +(5.56630 - 6.97992i) q^{74} +8.28215 q^{75} +(3.09660 - 3.88301i) q^{76} +(-3.94935 + 1.90191i) q^{77} +(6.20651 + 7.78272i) q^{78} +(-5.86220 - 7.35096i) q^{79} +(0.445708 + 0.214642i) q^{80} +(2.02476 - 8.87107i) q^{81} +(0.892813 + 3.91167i) q^{82} +(-11.4508 + 5.51441i) q^{83} +(4.92476 + 2.37164i) q^{84} +(0.585409 + 2.56484i) q^{85} -1.39672 q^{86} +(8.41880 + 4.13445i) q^{87} +1.39672 q^{88} +(-0.398796 - 1.74724i) q^{89} +(0.0149052 + 0.00717798i) q^{90} +(16.1610 - 7.78272i) q^{91} +(0.181045 + 0.793210i) q^{92} +(2.46615 - 10.8049i) q^{93} +(9.27831 + 4.46820i) q^{94} +(-1.53188 - 1.92092i) q^{95} +(-1.08592 - 1.36170i) q^{96} +(-3.42035 + 1.64715i) q^{97} +(1.77665 - 2.22785i) q^{98} +0.0467086 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 \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).

\(n\) \(31\)
\(\chi(n)\) \(e\left(\frac{3}{7}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.222521 − 0.974928i −0.157346 − 0.689378i
\(3\) 1.56920 + 0.755686i 0.905977 + 0.436295i 0.828044 − 0.560663i \(-0.189454\pi\)
0.0779327 + 0.996959i \(0.475168\pi\)
\(4\) −0.900969 + 0.433884i −0.450484 + 0.216942i
\(5\) 0.110081 + 0.482295i 0.0492296 + 0.215689i 0.993560 − 0.113311i \(-0.0361456\pi\)
−0.944330 + 0.329000i \(0.893288\pi\)
\(6\) 0.387560 − 1.69801i 0.158221 − 0.693210i
\(7\) −2.82760 − 1.36170i −1.06873 − 0.514674i −0.185033 − 0.982732i \(-0.559239\pi\)
−0.883697 + 0.468059i \(0.844954\pi\)
\(8\) 0.623490 + 0.781831i 0.220437 + 0.276419i
\(9\) 0.0208506 + 0.0261458i 0.00695019 + 0.00871526i
\(10\) 0.445708 − 0.214642i 0.140945 − 0.0678756i
\(11\) 0.870839 − 1.09200i 0.262568 − 0.329250i −0.633019 − 0.774136i \(-0.718185\pi\)
0.895587 + 0.444887i \(0.146756\pi\)
\(12\) −1.74168 −0.502779
\(13\) −3.56353 + 4.46852i −0.988344 + 1.23934i −0.0174472 + 0.999848i \(0.505554\pi\)
−0.970897 + 0.239497i \(0.923018\pi\)
\(14\) −0.698359 + 3.05971i −0.186644 + 0.817742i
\(15\) −0.191725 + 0.840003i −0.0495032 + 0.216888i
\(16\) 0.623490 − 0.781831i 0.155872 − 0.195458i
\(17\) 5.31800 1.28980 0.644902 − 0.764265i \(-0.276898\pi\)
0.644902 + 0.764265i \(0.276898\pi\)
\(18\) 0.0208506 − 0.0261458i 0.00491453 − 0.00616262i
\(19\) −4.47471 + 2.15491i −1.02657 + 0.494370i −0.869872 − 0.493277i \(-0.835799\pi\)
−0.156697 + 0.987647i \(0.550085\pi\)
\(20\) −0.308439 − 0.386771i −0.0689691 − 0.0864845i
\(21\) −3.40804 − 4.27355i −0.743695 − 0.932565i
\(22\) −1.25840 − 0.606013i −0.268292 − 0.129202i
\(23\) 0.181045 − 0.793210i 0.0377505 − 0.165396i −0.952539 − 0.304417i \(-0.901538\pi\)
0.990289 + 0.139021i \(0.0443956\pi\)
\(24\) 0.387560 + 1.69801i 0.0791103 + 0.346605i
\(25\) 4.28435 − 2.06324i 0.856871 − 0.412647i
\(26\) 5.14944 + 2.47984i 1.00989 + 0.486337i
\(27\) −1.14972 − 5.03725i −0.221263 − 0.969419i
\(28\) 3.13840 0.593101
\(29\) 5.38501 + 0.0414712i 0.999970 + 0.00770101i
\(30\) 0.861605 0.157307
\(31\) −1.41596 − 6.20373i −0.254314 − 1.11422i −0.927226 − 0.374501i \(-0.877814\pi\)
0.672912 − 0.739722i \(-0.265043\pi\)
\(32\) −0.900969 − 0.433884i −0.159270 − 0.0767005i
\(33\) 2.19173 − 1.05548i 0.381530 − 0.183735i
\(34\) −1.18337 − 5.18466i −0.202946 − 0.889162i
\(35\) 0.345477 − 1.51363i 0.0583962 − 0.255851i
\(36\) −0.0301299 − 0.0145098i −0.00502166 − 0.00241830i
\(37\) 5.56630 + 6.97992i 0.915095 + 1.14749i 0.988655 + 0.150202i \(0.0479923\pi\)
−0.0735606 + 0.997291i \(0.523436\pi\)
\(38\) 3.09660 + 3.88301i 0.502335 + 0.629908i
\(39\) −8.96867 + 4.31909i −1.43614 + 0.691607i
\(40\) −0.308439 + 0.386771i −0.0487685 + 0.0611538i
\(41\) −4.01226 −0.626610 −0.313305 − 0.949652i \(-0.601436\pi\)
−0.313305 + 0.949652i \(0.601436\pi\)
\(42\) −3.40804 + 4.27355i −0.525872 + 0.659423i
\(43\) 0.310799 − 1.36170i 0.0473964 − 0.207657i −0.945685 − 0.325084i \(-0.894607\pi\)
0.993082 + 0.117427i \(0.0374646\pi\)
\(44\) −0.310799 + 1.36170i −0.0468547 + 0.205284i
\(45\) −0.0103147 + 0.0129343i −0.00153763 + 0.00192813i
\(46\) −0.813609 −0.119960
\(47\) −6.42079 + 8.05142i −0.936569 + 1.17442i 0.0478986 + 0.998852i \(0.484748\pi\)
−0.984467 + 0.175568i \(0.943824\pi\)
\(48\) 1.56920 − 0.755686i 0.226494 − 0.109074i
\(49\) 1.77665 + 2.22785i 0.253807 + 0.318264i
\(50\) −2.96486 − 3.71782i −0.419295 − 0.525780i
\(51\) 8.34499 + 4.01873i 1.16853 + 0.562735i
\(52\) 1.27181 − 5.57215i 0.176368 − 0.772719i
\(53\) −0.944288 − 4.13720i −0.129708 − 0.568288i −0.997456 − 0.0712835i \(-0.977291\pi\)
0.867748 − 0.497004i \(-0.165567\pi\)
\(54\) −4.65512 + 2.24179i −0.633481 + 0.305068i
\(55\) 0.622528 + 0.299794i 0.0839416 + 0.0404241i
\(56\) −0.698359 − 3.05971i −0.0933221 − 0.408871i
\(57\) −8.65014 −1.14574
\(58\) −1.15784 − 5.25922i −0.152032 − 0.690569i
\(59\) 11.1598 1.45288 0.726438 − 0.687232i \(-0.241174\pi\)
0.726438 + 0.687232i \(0.241174\pi\)
\(60\) −0.191725 − 0.840003i −0.0247516 − 0.108444i
\(61\) −4.38454 − 2.11149i −0.561383 − 0.270348i 0.131598 − 0.991303i \(-0.457989\pi\)
−0.692982 + 0.720955i \(0.743703\pi\)
\(62\) −5.73311 + 2.76092i −0.728106 + 0.350637i
\(63\) −0.0233543 − 0.102322i −0.00294237 − 0.0128913i
\(64\) −0.222521 + 0.974928i −0.0278151 + 0.121866i
\(65\) −2.54742 − 1.22677i −0.315969 − 0.152163i
\(66\) −1.51672 − 1.90191i −0.186695 − 0.234109i
\(67\) 4.45072 + 5.58102i 0.543742 + 0.681830i 0.975460 − 0.220178i \(-0.0706639\pi\)
−0.431718 + 0.902009i \(0.642092\pi\)
\(68\) −4.79135 + 2.30739i −0.581036 + 0.279812i
\(69\) 0.883513 − 1.10789i 0.106362 − 0.133374i
\(70\) −1.55256 −0.185566
\(71\) 3.76423 − 4.72019i 0.446732 − 0.560184i −0.506572 − 0.862198i \(-0.669087\pi\)
0.953304 + 0.302014i \(0.0976589\pi\)
\(72\) −0.00744148 + 0.0326033i −0.000876987 + 0.00384233i
\(73\) −2.73238 + 11.9713i −0.319801 + 1.40114i 0.518103 + 0.855319i \(0.326639\pi\)
−0.837903 + 0.545819i \(0.816219\pi\)
\(74\) 5.56630 − 6.97992i 0.647070 − 0.811400i
\(75\) 8.28215 0.956341
\(76\) 3.09660 − 3.88301i 0.355204 − 0.445412i
\(77\) −3.94935 + 1.90191i −0.450070 + 0.216743i
\(78\) 6.20651 + 7.78272i 0.702749 + 0.881220i
\(79\) −5.86220 − 7.35096i −0.659549 − 0.827048i 0.333745 − 0.942663i \(-0.391687\pi\)
−0.993294 + 0.115615i \(0.963116\pi\)
\(80\) 0.445708 + 0.214642i 0.0498316 + 0.0239977i
\(81\) 2.02476 − 8.87107i 0.224974 − 0.985675i
\(82\) 0.892813 + 3.91167i 0.0985947 + 0.431971i
\(83\) −11.4508 + 5.51441i −1.25689 + 0.605285i −0.939350 − 0.342960i \(-0.888570\pi\)
−0.317538 + 0.948246i \(0.602856\pi\)
\(84\) 4.92476 + 2.37164i 0.537336 + 0.258767i
\(85\) 0.585409 + 2.56484i 0.0634965 + 0.278196i
\(86\) −1.39672 −0.150612
\(87\) 8.41880 + 4.13445i 0.902590 + 0.443259i
\(88\) 1.39672 0.148891
\(89\) −0.398796 − 1.74724i −0.0422722 − 0.185207i 0.949384 − 0.314118i \(-0.101709\pi\)
−0.991656 + 0.128911i \(0.958852\pi\)
\(90\) 0.0149052 + 0.00717798i 0.00157115 + 0.000756626i
\(91\) 16.1610 − 7.78272i 1.69413 − 0.815851i
\(92\) 0.181045 + 0.793210i 0.0188753 + 0.0826979i
\(93\) 2.46615 − 10.8049i 0.255728 − 1.12042i
\(94\) 9.27831 + 4.46820i 0.956985 + 0.460860i
\(95\) −1.53188 − 1.92092i −0.157168 − 0.197082i
\(96\) −1.08592 − 1.36170i −0.110831 − 0.138978i
\(97\) −3.42035 + 1.64715i −0.347283 + 0.167243i −0.599391 − 0.800456i \(-0.704591\pi\)
0.252108 + 0.967699i \(0.418876\pi\)
\(98\) 1.77665 − 2.22785i 0.179469 − 0.225047i
\(99\) 0.0467086 0.00469439
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.7.2 ✓ 12
3.2 odd 2 522.2.k.h.181.1 12
4.3 odd 2 464.2.u.h.65.1 12
29.2 odd 28 1682.2.b.i.1681.5 12
29.5 even 14 1682.2.a.q.1.5 6
29.24 even 7 1682.2.a.t.1.2 6
29.25 even 7 inner 58.2.d.b.25.2 yes 12
29.27 odd 28 1682.2.b.i.1681.8 12
87.83 odd 14 522.2.k.h.199.1 12
116.83 odd 14 464.2.u.h.257.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.7.2 ✓ 12 1.1 even 1 trivial
58.2.d.b.25.2 yes 12 29.25 even 7 inner
464.2.u.h.65.1 12 4.3 odd 2
464.2.u.h.257.1 12 116.83 odd 14
522.2.k.h.181.1 12 3.2 odd 2
522.2.k.h.199.1 12 87.83 odd 14
1682.2.a.q.1.5 6 29.5 even 14
1682.2.a.t.1.2 6 29.24 even 7
1682.2.b.i.1681.5 12 29.2 odd 28
1682.2.b.i.1681.8 12 29.27 odd 28