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Properties

Label 58.2.d.b.53.2
Level $58$
Weight $2$
Character 58.53
Analytic conductor $0.463$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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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 53.2
Root \(-0.260453 + 1.14112i\) of defining polynomial
Character \(\chi\) \(=\) 58.53
Dual form 58.2.d.b.23.2

$q$-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+(0.623490 + 0.781831i) q^{2} +(0.260453 + 1.14112i) q^{3} +(-0.222521 + 0.974928i) q^{4} +(-1.85326 - 2.32392i) q^{5} +(-0.729773 + 0.915107i) q^{6} +(-0.115912 - 0.507846i) q^{7} +(-0.900969 + 0.433884i) q^{8} +(1.46859 - 0.707235i) q^{9} +(0.661422 - 2.89788i) q^{10} +(0.585233 + 0.281833i) q^{11} -1.17047 q^{12} +(-0.444717 - 0.214164i) q^{13} +(0.324780 - 0.407261i) q^{14} +(2.16918 - 2.72007i) q^{15} +(-0.900969 - 0.433884i) q^{16} -7.42032 q^{17} +(1.46859 + 0.707235i) q^{18} +(-1.47532 + 6.46378i) q^{19} +(2.67804 - 1.28968i) q^{20} +(0.549323 - 0.264540i) q^{21} +(0.144541 + 0.633273i) q^{22} +(4.74970 - 5.95594i) q^{23} +(-0.729773 - 0.915107i) q^{24} +(-0.853408 + 3.73902i) q^{25} +(-0.109836 - 0.481223i) q^{26} +(3.37886 + 4.23695i) q^{27} +0.520906 q^{28} +(-4.56917 + 2.85003i) q^{29} +3.47909 q^{30} +(3.72875 + 4.67571i) q^{31} +(-0.222521 - 0.974928i) q^{32} +(-0.169180 + 0.741224i) q^{33} +(-4.62649 - 5.80144i) q^{34} +(-0.965376 + 1.21054i) q^{35} +(0.362712 + 1.58914i) q^{36} +(2.23650 - 1.07704i) q^{37} +(-5.97343 + 2.87665i) q^{38} +(0.128559 - 0.563254i) q^{39} +(2.67804 + 1.28968i) q^{40} +7.82245 q^{41} +(0.549323 + 0.264540i) q^{42} +(0.404994 - 0.507846i) q^{43} +(-0.404994 + 0.507846i) q^{44} +(-4.36524 - 2.10219i) q^{45} +7.61793 q^{46} +(-7.92488 - 3.81642i) q^{47} +(0.260453 - 1.14112i) q^{48} +(6.06231 - 2.91945i) q^{49} +(-3.45538 + 1.66402i) q^{50} +(-1.93264 - 8.46747i) q^{51} +(0.307754 - 0.385911i) q^{52} +(-0.717766 - 0.900050i) q^{53} +(-1.20590 + 5.28339i) q^{54} +(-0.429633 - 1.88234i) q^{55} +(0.324780 + 0.407261i) q^{56} -7.76020 q^{57} +(-5.07708 - 1.79536i) q^{58} -5.31686 q^{59} +(2.16918 + 2.72007i) q^{60} +(-2.15779 - 9.45389i) q^{61} +(-1.33078 + 5.83051i) q^{62} +(-0.529394 - 0.663840i) q^{63} +(0.623490 - 0.781831i) q^{64} +(0.326477 + 1.43039i) q^{65} +(-0.684994 + 0.329876i) q^{66} +(-4.07320 + 1.96155i) q^{67} +(1.65118 - 7.23427i) q^{68} +(8.03351 + 3.86873i) q^{69} -1.54834 q^{70} +(12.7440 + 6.13719i) q^{71} +(-1.01630 + 1.27439i) q^{72} +(5.44958 - 6.83356i) q^{73} +(2.23650 + 1.07704i) q^{74} -4.48894 q^{75} +(-5.97343 - 2.87665i) q^{76} +(0.0752920 - 0.329876i) q^{77} +(0.520525 - 0.250672i) q^{78} +(-3.71962 + 1.79127i) q^{79} +(0.661422 + 2.89788i) q^{80} +(-0.905949 + 1.13602i) q^{81} +(4.87722 + 6.11584i) q^{82} +(-0.952729 + 4.17418i) q^{83} +(0.135672 + 0.594416i) q^{84} +(13.7518 + 17.2442i) q^{85} +0.649559 q^{86} +(-4.44228 - 4.47167i) q^{87} -0.649559 q^{88} +(-0.743610 - 0.932457i) q^{89} +(-1.07813 - 4.72357i) q^{90} +(-0.0572142 + 0.250672i) q^{91} +(4.74970 + 5.95594i) q^{92} +(-4.36437 + 5.47275i) q^{93} +(-1.95728 - 8.57542i) q^{94} +(17.7555 - 8.55057i) q^{95} +(1.05455 - 0.507846i) q^{96} +(-2.61081 + 11.4387i) q^{97} +(6.06231 + 2.91945i) q^{98} +1.05879 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 \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).

\(n\) \(31\)
\(\chi(n)\) \(e\left(\frac{2}{7}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.623490 + 0.781831i 0.440874 + 0.552838i
\(3\) 0.260453 + 1.14112i 0.150373 + 0.658825i 0.992776 + 0.119979i \(0.0382826\pi\)
−0.842404 + 0.538847i \(0.818860\pi\)
\(4\) −0.222521 + 0.974928i −0.111260 + 0.487464i
\(5\) −1.85326 − 2.32392i −0.828804 − 1.03929i −0.998552 − 0.0538002i \(-0.982867\pi\)
0.169747 − 0.985488i \(-0.445705\pi\)
\(6\) −0.729773 + 0.915107i −0.297929 + 0.373591i
\(7\) −0.115912 − 0.507846i −0.0438108 − 0.191948i 0.948287 − 0.317414i \(-0.102814\pi\)
−0.992098 + 0.125466i \(0.959957\pi\)
\(8\) −0.900969 + 0.433884i −0.318541 + 0.153401i
\(9\) 1.46859 − 0.707235i 0.489530 − 0.235745i
\(10\) 0.661422 − 2.89788i 0.209160 − 0.916390i
\(11\) 0.585233 + 0.281833i 0.176454 + 0.0849759i 0.520026 − 0.854151i \(-0.325922\pi\)
−0.343571 + 0.939127i \(0.611637\pi\)
\(12\) −1.17047 −0.337884
\(13\) −0.444717 − 0.214164i −0.123342 − 0.0593985i 0.371195 − 0.928555i \(-0.378948\pi\)
−0.494537 + 0.869156i \(0.664662\pi\)
\(14\) 0.324780 − 0.407261i 0.0868010 − 0.108845i
\(15\) 2.16918 − 2.72007i 0.560080 − 0.702318i
\(16\) −0.900969 − 0.433884i −0.225242 − 0.108471i
\(17\) −7.42032 −1.79969 −0.899846 − 0.436208i \(-0.856321\pi\)
−0.899846 + 0.436208i \(0.856321\pi\)
\(18\) 1.46859 + 0.707235i 0.346150 + 0.166697i
\(19\) −1.47532 + 6.46378i −0.338461 + 1.48289i 0.463811 + 0.885934i \(0.346482\pi\)
−0.802272 + 0.596959i \(0.796375\pi\)
\(20\) 2.67804 − 1.28968i 0.598828 − 0.288381i
\(21\) 0.549323 − 0.264540i 0.119872 − 0.0577273i
\(22\) 0.144541 + 0.633273i 0.0308161 + 0.135014i
\(23\) 4.74970 − 5.95594i 0.990381 − 1.24190i 0.0201303 − 0.999797i \(-0.493592\pi\)
0.970251 − 0.242101i \(-0.0778367\pi\)
\(24\) −0.729773 − 0.915107i −0.148964 − 0.186795i
\(25\) −0.853408 + 3.73902i −0.170682 + 0.747805i
\(26\) −0.109836 − 0.481223i −0.0215406 − 0.0943755i
\(27\) 3.37886 + 4.23695i 0.650261 + 0.815402i
\(28\) 0.520906 0.0984420
\(29\) −4.56917 + 2.85003i −0.848474 + 0.529237i
\(30\) 3.47909 0.635193
\(31\) 3.72875 + 4.67571i 0.669703 + 0.839782i 0.994361 − 0.106050i \(-0.0338204\pi\)
−0.324657 + 0.945832i \(0.605249\pi\)
\(32\) −0.222521 − 0.974928i −0.0393365 − 0.172345i
\(33\) −0.169180 + 0.741224i −0.0294504 + 0.129031i
\(34\) −4.62649 − 5.80144i −0.793437 − 0.994938i
\(35\) −0.965376 + 1.21054i −0.163178 + 0.204619i
\(36\) 0.362712 + 1.58914i 0.0604519 + 0.264857i
\(37\) 2.23650 − 1.07704i 0.367678 − 0.177065i −0.240916 − 0.970546i \(-0.577448\pi\)
0.608595 + 0.793481i \(0.291734\pi\)
\(38\) −5.97343 + 2.87665i −0.969019 + 0.466655i
\(39\) 0.128559 − 0.563254i 0.0205859 − 0.0901929i
\(40\) 2.67804 + 1.28968i 0.423436 + 0.203916i
\(41\) 7.82245 1.22166 0.610830 − 0.791761i \(-0.290836\pi\)
0.610830 + 0.791761i \(0.290836\pi\)
\(42\) 0.549323 + 0.264540i 0.0847623 + 0.0408194i
\(43\) 0.404994 − 0.507846i 0.0617609 − 0.0774458i −0.749991 − 0.661447i \(-0.769942\pi\)
0.811752 + 0.584002i \(0.198514\pi\)
\(44\) −0.404994 + 0.507846i −0.0610551 + 0.0765606i
\(45\) −4.36524 − 2.10219i −0.650731 − 0.313376i
\(46\) 7.61793 1.12320
\(47\) −7.92488 − 3.81642i −1.15596 − 0.556682i −0.245143 − 0.969487i \(-0.578835\pi\)
−0.910819 + 0.412805i \(0.864549\pi\)
\(48\) 0.260453 − 1.14112i 0.0375932 − 0.164706i
\(49\) 6.06231 − 2.91945i 0.866044 − 0.417065i
\(50\) −3.45538 + 1.66402i −0.488664 + 0.235328i
\(51\) −1.93264 − 8.46747i −0.270624 − 1.18568i
\(52\) 0.307754 − 0.385911i 0.0426777 − 0.0535162i
\(53\) −0.717766 − 0.900050i −0.0985927 − 0.123631i 0.730088 − 0.683354i \(-0.239479\pi\)
−0.828680 + 0.559722i \(0.810908\pi\)
\(54\) −1.20590 + 5.28339i −0.164102 + 0.718979i
\(55\) −0.429633 − 1.88234i −0.0579317 − 0.253815i
\(56\) 0.324780 + 0.407261i 0.0434005 + 0.0544225i
\(57\) −7.76020 −1.02786
\(58\) −5.07708 − 1.79536i −0.666653 − 0.235742i
\(59\) −5.31686 −0.692196 −0.346098 − 0.938198i \(-0.612493\pi\)
−0.346098 + 0.938198i \(0.612493\pi\)
\(60\) 2.16918 + 2.72007i 0.280040 + 0.351159i
\(61\) −2.15779 − 9.45389i −0.276277 − 1.21045i −0.902461 − 0.430772i \(-0.858241\pi\)
0.626184 − 0.779675i \(-0.284616\pi\)
\(62\) −1.33078 + 5.83051i −0.169009 + 0.740475i
\(63\) −0.529394 − 0.663840i −0.0666974 − 0.0836359i
\(64\) 0.623490 − 0.781831i 0.0779362 − 0.0977289i
\(65\) 0.326477 + 1.43039i 0.0404945 + 0.177418i
\(66\) −0.684994 + 0.329876i −0.0843170 + 0.0406049i
\(67\) −4.07320 + 1.96155i −0.497620 + 0.239641i −0.665821 − 0.746112i \(-0.731919\pi\)
0.168201 + 0.985753i \(0.446204\pi\)
\(68\) 1.65118 − 7.23427i 0.200234 − 0.877285i
\(69\) 8.03351 + 3.86873i 0.967121 + 0.465741i
\(70\) −1.54834 −0.185062
\(71\) 12.7440 + 6.13719i 1.51243 + 0.728350i 0.992081 − 0.125601i \(-0.0400860\pi\)
0.520353 + 0.853951i \(0.325800\pi\)
\(72\) −1.01630 + 1.27439i −0.119772 + 0.150189i
\(73\) 5.44958 − 6.83356i 0.637825 − 0.799808i −0.352904 − 0.935660i \(-0.614806\pi\)
0.990729 + 0.135852i \(0.0433772\pi\)
\(74\) 2.23650 + 1.07704i 0.259988 + 0.125204i
\(75\) −4.48894 −0.518339
\(76\) −5.97343 − 2.87665i −0.685200 − 0.329975i
\(77\) 0.0752920 − 0.329876i 0.00858032 − 0.0375928i
\(78\) 0.520525 − 0.250672i 0.0589379 − 0.0283830i
\(79\) −3.71962 + 1.79127i −0.418490 + 0.201534i −0.631265 − 0.775567i \(-0.717464\pi\)
0.212775 + 0.977101i \(0.431750\pi\)
\(80\) 0.661422 + 2.89788i 0.0739492 + 0.323993i
\(81\) −0.905949 + 1.13602i −0.100661 + 0.126225i
\(82\) 4.87722 + 6.11584i 0.538598 + 0.675381i
\(83\) −0.952729 + 4.17418i −0.104576 + 0.458176i 0.895343 + 0.445378i \(0.146931\pi\)
−0.999918 + 0.0127974i \(0.995926\pi\)
\(84\) 0.135672 + 0.594416i 0.0148030 + 0.0648561i
\(85\) 13.7518 + 17.2442i 1.49159 + 1.87040i
\(86\) 0.649559 0.0700438
\(87\) −4.44228 − 4.47167i −0.476262 − 0.479413i
\(88\) −0.649559 −0.0692432
\(89\) −0.743610 − 0.932457i −0.0788225 − 0.0988403i 0.740857 − 0.671663i \(-0.234420\pi\)
−0.819679 + 0.572823i \(0.805848\pi\)
\(90\) −1.07813 − 4.72357i −0.113644 − 0.497909i
\(91\) −0.0572142 + 0.250672i −0.00599768 + 0.0262775i
\(92\) 4.74970 + 5.95594i 0.495191 + 0.620949i
\(93\) −4.36437 + 5.47275i −0.452564 + 0.567498i
\(94\) −1.95728 − 8.57542i −0.201878 − 0.884487i
\(95\) 17.7555 − 8.55057i 1.82167 − 0.877270i
\(96\) 1.05455 − 0.507846i 0.107630 − 0.0518318i
\(97\) −2.61081 + 11.4387i −0.265088 + 1.16143i 0.650563 + 0.759452i \(0.274533\pi\)
−0.915651 + 0.401974i \(0.868324\pi\)
\(98\) 6.06231 + 2.91945i 0.612386 + 0.294909i
\(99\) 1.05879 0.106412
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.53.2 yes 12
3.2 odd 2 522.2.k.h.343.2 12
4.3 odd 2 464.2.u.h.401.1 12
29.8 odd 28 1682.2.b.i.1681.9 12
29.9 even 14 1682.2.a.q.1.4 6
29.20 even 7 1682.2.a.t.1.3 6
29.21 odd 28 1682.2.b.i.1681.4 12
29.23 even 7 inner 58.2.d.b.23.2 ✓ 12
87.23 odd 14 522.2.k.h.487.2 12
116.23 odd 14 464.2.u.h.81.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.23.2 ✓ 12 29.23 even 7 inner
58.2.d.b.53.2 yes 12 1.1 even 1 trivial
464.2.u.h.81.1 12 116.23 odd 14
464.2.u.h.401.1 12 4.3 odd 2
522.2.k.h.343.2 12 3.2 odd 2
522.2.k.h.487.2 12 87.23 odd 14
1682.2.a.q.1.4 6 29.9 even 14
1682.2.a.t.1.3 6 29.20 even 7
1682.2.b.i.1681.4 12 29.21 odd 28
1682.2.b.i.1681.9 12 29.8 odd 28