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Properties

Label 58.2.d.a.53.1
Level $58$
Weight $2$
Character 58.53
Analytic conductor $0.463$
Analytic rank $0$
Dimension $6$
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 = [6] 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: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label 53.1
Root \(0.222521 - 0.974928i\) of defining polynomial
Character \(\chi\) \(=\) 58.53
Dual form 58.2.d.a.23.1

$q$-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+(-0.623490 - 0.781831i) q^{2} +(0.500000 + 2.19064i) q^{3} +(-0.222521 + 0.974928i) q^{4} +(0.969501 + 1.21572i) q^{5} +(1.40097 - 1.75676i) q^{6} +(-0.777479 - 3.40636i) q^{7} +(0.900969 - 0.433884i) q^{8} +(-1.84601 + 0.888992i) q^{9} +(0.346011 - 1.51597i) q^{10} +(-3.37047 - 1.62313i) q^{11} -2.24698 q^{12} +(3.46950 + 1.67082i) q^{13} +(-2.17845 + 2.73169i) q^{14} +(-2.17845 + 2.73169i) q^{15} +(-0.900969 - 0.433884i) q^{16} -4.93900 q^{17} +(1.84601 + 0.888992i) q^{18} +(1.37047 - 6.00442i) q^{19} +(-1.40097 + 0.674671i) q^{20} +(7.07338 - 3.40636i) q^{21} +(0.832437 + 3.64715i) q^{22} +(-3.59299 + 4.50547i) q^{23} +(1.40097 + 1.75676i) q^{24} +(0.574572 - 2.51737i) q^{25} +(-0.856896 - 3.75431i) q^{26} +(1.33244 + 1.67082i) q^{27} +3.49396 q^{28} +(5.38135 - 0.202542i) q^{29} +3.49396 q^{30} +(1.56853 + 1.96688i) q^{31} +(0.222521 + 0.974928i) q^{32} +(1.87047 - 8.19506i) q^{33} +(3.07942 + 3.86147i) q^{34} +(3.38740 - 4.24766i) q^{35} +(-0.455927 - 1.99755i) q^{36} +(-6.11745 + 2.94601i) q^{37} +(-5.54892 + 2.67222i) q^{38} +(-1.92543 + 8.43585i) q^{39} +(1.40097 + 0.674671i) q^{40} +0.0271471 q^{41} +(-7.07338 - 3.40636i) q^{42} +(-7.32036 + 9.17944i) q^{43} +(2.33244 - 2.92478i) q^{44} +(-2.87047 - 1.38235i) q^{45} +5.76271 q^{46} +(0.500000 + 0.240787i) q^{47} +(0.500000 - 2.19064i) q^{48} +(-4.69202 + 2.25956i) q^{49} +(-2.32640 + 1.12033i) q^{50} +(-2.46950 - 10.8196i) q^{51} +(-2.40097 + 3.01072i) q^{52} +(2.03199 + 2.54804i) q^{53} +(0.475541 - 2.08348i) q^{54} +(-1.29440 - 5.67116i) q^{55} +(-2.17845 - 2.73169i) q^{56} +13.8388 q^{57} +(-3.51357 - 4.08103i) q^{58} -3.06100 q^{59} +(-2.17845 - 2.73169i) q^{60} +(-0.0293030 - 0.128385i) q^{61} +(0.559802 - 2.45265i) q^{62} +(4.46346 + 5.59700i) q^{63} +(0.623490 - 0.781831i) q^{64} +(1.33244 + 5.83779i) q^{65} +(-7.57338 + 3.64715i) q^{66} +(9.91066 - 4.77272i) q^{67} +(1.09903 - 4.81517i) q^{68} +(-11.6664 - 5.61823i) q^{69} -5.43296 q^{70} +(3.26659 + 1.57311i) q^{71} +(-1.27748 + 1.60191i) q^{72} +(-4.13102 + 5.18014i) q^{73} +(6.11745 + 2.94601i) q^{74} +5.80194 q^{75} +(5.54892 + 2.67222i) q^{76} +(-2.90850 + 12.7430i) q^{77} +(7.79590 - 3.75431i) q^{78} +(-1.35690 + 0.653447i) q^{79} +(-0.346011 - 1.51597i) q^{80} +(-6.82640 + 8.56003i) q^{81} +(-0.0169259 - 0.0212244i) q^{82} +(2.21768 - 9.71628i) q^{83} +(1.74698 + 7.65402i) q^{84} +(-4.78836 - 6.00442i) q^{85} +11.7409 q^{86} +(3.13437 + 11.6874i) q^{87} -3.74094 q^{88} +(-3.36443 - 4.21886i) q^{89} +(0.708947 + 3.10610i) q^{90} +(2.99396 - 13.1174i) q^{91} +(-3.59299 - 4.50547i) q^{92} +(-3.52446 + 4.41953i) q^{93} +(-0.123490 - 0.541044i) q^{94} +(8.62833 - 4.15519i) q^{95} +(-2.02446 + 0.974928i) q^{96} +(-0.960771 + 4.20941i) q^{97} +(4.69202 + 2.25956i) q^{98} +7.66487 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} + 3 q^{3} - q^{4} - 4 q^{5} + 4 q^{6} - 5 q^{7} + q^{8} - 6 q^{9} - 3 q^{10} - 6 q^{11} - 4 q^{12} + 11 q^{13} - 9 q^{14} - 9 q^{15} - q^{16} - 10 q^{17} + 6 q^{18} - 6 q^{19} - 4 q^{20}+ \cdots + 48 q^{99}+O(q^{100}) \) 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
\(3\) 0.500000 + 2.19064i 0.288675 + 1.26477i 0.886345 + 0.463026i \(0.153236\pi\)
−0.597670 + 0.801742i \(0.703907\pi\)
\(4\) −0.222521 + 0.974928i −0.111260 + 0.487464i
\(5\) 0.969501 + 1.21572i 0.433574 + 0.543684i 0.949837 − 0.312746i \(-0.101249\pi\)
−0.516263 + 0.856430i \(0.672677\pi\)
\(6\) 1.40097 − 1.75676i 0.571943 − 0.717194i
\(7\) −0.777479 − 3.40636i −0.293859 − 1.28748i −0.879106 − 0.476627i \(-0.841859\pi\)
0.585246 − 0.810856i \(-0.300998\pi\)
\(8\) 0.900969 − 0.433884i 0.318541 − 0.153401i
\(9\) −1.84601 + 0.888992i −0.615337 + 0.296331i
\(10\) 0.346011 − 1.51597i 0.109418 − 0.479392i
\(11\) −3.37047 − 1.62313i −1.01623 − 0.489393i −0.149818 − 0.988714i \(-0.547869\pi\)
−0.866417 + 0.499321i \(0.833583\pi\)
\(12\) −2.24698 −0.648647
\(13\) 3.46950 + 1.67082i 0.962266 + 0.463403i 0.847970 − 0.530044i \(-0.177824\pi\)
0.114296 + 0.993447i \(0.463539\pi\)
\(14\) −2.17845 + 2.73169i −0.582215 + 0.730074i
\(15\) −2.17845 + 2.73169i −0.562473 + 0.705319i
\(16\) −0.900969 − 0.433884i −0.225242 − 0.108471i
\(17\) −4.93900 −1.19788 −0.598942 − 0.800793i \(-0.704412\pi\)
−0.598942 + 0.800793i \(0.704412\pi\)
\(18\) 1.84601 + 0.888992i 0.435109 + 0.209537i
\(19\) 1.37047 − 6.00442i 0.314407 − 1.37751i −0.532798 − 0.846242i \(-0.678859\pi\)
0.847205 − 0.531266i \(-0.178283\pi\)
\(20\) −1.40097 + 0.674671i −0.313266 + 0.150861i
\(21\) 7.07338 − 3.40636i 1.54354 − 0.743328i
\(22\) 0.832437 + 3.64715i 0.177476 + 0.777574i
\(23\) −3.59299 + 4.50547i −0.749190 + 0.939455i −0.999588 − 0.0286881i \(-0.990867\pi\)
0.250398 + 0.968143i \(0.419438\pi\)
\(24\) 1.40097 + 1.75676i 0.285972 + 0.358597i
\(25\) 0.574572 − 2.51737i 0.114914 − 0.503473i
\(26\) −0.856896 − 3.75431i −0.168051 − 0.736280i
\(27\) 1.33244 + 1.67082i 0.256428 + 0.321550i
\(28\) 3.49396 0.660296
\(29\) 5.38135 − 0.202542i 0.999292 − 0.0376111i
\(30\) 3.49396 0.637907
\(31\) 1.56853 + 1.96688i 0.281717 + 0.353261i 0.902476 − 0.430739i \(-0.141747\pi\)
−0.620760 + 0.784001i \(0.713176\pi\)
\(32\) 0.222521 + 0.974928i 0.0393365 + 0.172345i
\(33\) 1.87047 − 8.19506i 0.325607 − 1.42658i
\(34\) 3.07942 + 3.86147i 0.528116 + 0.662236i
\(35\) 3.38740 − 4.24766i 0.572574 − 0.717985i
\(36\) −0.455927 − 1.99755i −0.0759878 − 0.332924i
\(37\) −6.11745 + 2.94601i −1.00570 + 0.484321i −0.862870 − 0.505426i \(-0.831335\pi\)
−0.142833 + 0.989747i \(0.545621\pi\)
\(38\) −5.54892 + 2.67222i −0.900153 + 0.433491i
\(39\) −1.92543 + 8.43585i −0.308315 + 1.35082i
\(40\) 1.40097 + 0.674671i 0.221513 + 0.106675i
\(41\) 0.0271471 0.00423966 0.00211983 − 0.999998i \(-0.499325\pi\)
0.00211983 + 0.999998i \(0.499325\pi\)
\(42\) −7.07338 − 3.40636i −1.09145 − 0.525613i
\(43\) −7.32036 + 9.17944i −1.11634 + 1.39985i −0.209797 + 0.977745i \(0.567280\pi\)
−0.906547 + 0.422106i \(0.861291\pi\)
\(44\) 2.33244 − 2.92478i 0.351628 − 0.440928i
\(45\) −2.87047 − 1.38235i −0.427904 − 0.206068i
\(46\) 5.76271 0.849665
\(47\) 0.500000 + 0.240787i 0.0729325 + 0.0351224i 0.469994 − 0.882669i \(-0.344256\pi\)
−0.397062 + 0.917792i \(0.629970\pi\)
\(48\) 0.500000 − 2.19064i 0.0721688 − 0.316192i
\(49\) −4.69202 + 2.25956i −0.670289 + 0.322794i
\(50\) −2.32640 + 1.12033i −0.329002 + 0.158439i
\(51\) −2.46950 − 10.8196i −0.345799 − 1.51505i
\(52\) −2.40097 + 3.01072i −0.332954 + 0.417512i
\(53\) 2.03199 + 2.54804i 0.279115 + 0.350000i 0.901552 − 0.432670i \(-0.142429\pi\)
−0.622437 + 0.782670i \(0.713857\pi\)
\(54\) 0.475541 − 2.08348i 0.0647130 − 0.283526i
\(55\) −1.29440 − 5.67116i −0.174538 − 0.764699i
\(56\) −2.17845 − 2.73169i −0.291107 − 0.365037i
\(57\) 13.8388 1.83299
\(58\) −3.51357 − 4.08103i −0.461355 − 0.535865i
\(59\) −3.06100 −0.398508 −0.199254 − 0.979948i \(-0.563852\pi\)
−0.199254 + 0.979948i \(0.563852\pi\)
\(60\) −2.17845 − 2.73169i −0.281236 − 0.352659i
\(61\) −0.0293030 − 0.128385i −0.00375187 − 0.0164380i 0.973017 − 0.230732i \(-0.0741121\pi\)
−0.976769 + 0.214294i \(0.931255\pi\)
\(62\) 0.559802 − 2.45265i 0.0710950 − 0.311487i
\(63\) 4.46346 + 5.59700i 0.562343 + 0.705156i
\(64\) 0.623490 − 0.781831i 0.0779362 − 0.0977289i
\(65\) 1.33244 + 5.83779i 0.165268 + 0.724089i
\(66\) −7.57338 + 3.64715i −0.932218 + 0.448933i
\(67\) 9.91066 − 4.77272i 1.21078 − 0.583081i 0.284050 − 0.958810i \(-0.408322\pi\)
0.926730 + 0.375729i \(0.122608\pi\)
\(68\) 1.09903 − 4.81517i 0.133277 − 0.583925i
\(69\) −11.6664 − 5.61823i −1.40447 − 0.676355i
\(70\) −5.43296 −0.649363
\(71\) 3.26659 + 1.57311i 0.387673 + 0.186694i 0.617560 − 0.786524i \(-0.288121\pi\)
−0.229887 + 0.973217i \(0.573836\pi\)
\(72\) −1.27748 + 1.60191i −0.150552 + 0.188787i
\(73\) −4.13102 + 5.18014i −0.483500 + 0.606289i −0.962419 − 0.271570i \(-0.912457\pi\)
0.478919 + 0.877859i \(0.341029\pi\)
\(74\) 6.11745 + 2.94601i 0.711139 + 0.342466i
\(75\) 5.80194 0.669950
\(76\) 5.54892 + 2.67222i 0.636504 + 0.306524i
\(77\) −2.90850 + 12.7430i −0.331455 + 1.45220i
\(78\) 7.79590 − 3.75431i 0.882712 − 0.425091i
\(79\) −1.35690 + 0.653447i −0.152663 + 0.0735185i −0.508655 − 0.860970i \(-0.669857\pi\)
0.355992 + 0.934489i \(0.384143\pi\)
\(80\) −0.346011 − 1.51597i −0.0386852 − 0.169491i
\(81\) −6.82640 + 8.56003i −0.758488 + 0.951114i
\(82\) −0.0169259 − 0.0212244i −0.00186916 − 0.00234385i
\(83\) 2.21768 − 9.71628i 0.243422 − 1.06650i −0.694456 − 0.719535i \(-0.744355\pi\)
0.937878 − 0.346965i \(-0.112788\pi\)
\(84\) 1.74698 + 7.65402i 0.190611 + 0.835122i
\(85\) −4.78836 − 6.00442i −0.519371 − 0.651271i
\(86\) 11.7409 1.26606
\(87\) 3.13437 + 11.6874i 0.336040 + 1.25302i
\(88\) −3.74094 −0.398785
\(89\) −3.36443 − 4.21886i −0.356629 − 0.447198i 0.570861 − 0.821047i \(-0.306609\pi\)
−0.927490 + 0.373848i \(0.878038\pi\)
\(90\) 0.708947 + 3.10610i 0.0747296 + 0.327412i
\(91\) 2.99396 − 13.1174i 0.313852 − 1.37508i
\(92\) −3.59299 − 4.50547i −0.374595 − 0.469727i
\(93\) −3.52446 + 4.41953i −0.365469 + 0.458284i
\(94\) −0.123490 − 0.541044i −0.0127370 − 0.0558044i
\(95\) 8.62833 − 4.15519i 0.885248 − 0.426313i
\(96\) −2.02446 + 0.974928i −0.206620 + 0.0995032i
\(97\) −0.960771 + 4.20941i −0.0975515 + 0.427401i −0.999994 − 0.00340876i \(-0.998915\pi\)
0.902443 + 0.430810i \(0.141772\pi\)
\(98\) 4.69202 + 2.25956i 0.473966 + 0.228250i
\(99\) 7.66487 0.770349
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.a.53.1 yes 6
3.2 odd 2 522.2.k.c.343.1 6
4.3 odd 2 464.2.u.b.401.1 6
29.8 odd 28 1682.2.b.g.1681.1 6
29.9 even 14 1682.2.a.n.1.3 3
29.20 even 7 1682.2.a.m.1.1 3
29.21 odd 28 1682.2.b.g.1681.6 6
29.23 even 7 inner 58.2.d.a.23.1 ✓ 6
87.23 odd 14 522.2.k.c.487.1 6
116.23 odd 14 464.2.u.b.81.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.a.23.1 ✓ 6 29.23 even 7 inner
58.2.d.a.53.1 yes 6 1.1 even 1 trivial
464.2.u.b.81.1 6 116.23 odd 14
464.2.u.b.401.1 6 4.3 odd 2
522.2.k.c.343.1 6 3.2 odd 2
522.2.k.c.487.1 6 87.23 odd 14
1682.2.a.m.1.1 3 29.20 even 7
1682.2.a.n.1.3 3 29.9 even 14
1682.2.b.g.1681.1 6 29.8 odd 28
1682.2.b.g.1681.6 6 29.21 odd 28