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Properties

Label 58.2.d.b.45.2
Level $58$
Weight $2$
Character 58.45
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 45.2
Root \(-1.02179 - 1.28129i\) of defining polynomial
Character \(\chi\) \(=\) 58.45
Dual form 58.2.d.b.49.2

$q$-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+(-0.900969 - 0.433884i) q^{2} +(1.02179 - 1.28129i) q^{3} +(0.623490 + 0.781831i) q^{4} +(-1.07557 - 0.517965i) q^{5} +(-1.47653 + 0.711061i) q^{6} +(1.27416 - 1.59774i) q^{7} +(-0.222521 - 0.974928i) q^{8} +(0.0699247 + 0.306360i) q^{9} +(0.744314 + 0.933340i) q^{10} +(-0.819415 + 3.59009i) q^{11} +1.63883 q^{12} +(-0.479858 + 2.10239i) q^{13} +(-1.84121 + 0.886679i) q^{14} +(-1.76267 + 0.848856i) q^{15} +(-0.222521 + 0.974928i) q^{16} -6.53517 q^{17} +(0.0699247 - 0.306360i) q^{18} +(3.31337 + 4.15484i) q^{19} +(-0.265643 - 1.16386i) q^{20} +(-0.745242 - 3.26512i) q^{21} +(2.29595 - 2.87903i) q^{22} +(5.30987 - 2.55710i) q^{23} +(-1.47653 - 0.711061i) q^{24} +(-2.22890 - 2.79495i) q^{25} +(1.34453 - 1.68599i) q^{26} +(4.89359 + 2.35663i) q^{27} +2.04359 q^{28} +(1.75040 - 5.09275i) q^{29} +1.95641 q^{30} +(-8.10571 - 3.90350i) q^{31} +(0.623490 - 0.781831i) q^{32} +(3.76267 + 4.71824i) q^{33} +(5.88799 + 2.83551i) q^{34} +(-2.19801 + 1.05851i) q^{35} +(-0.195925 + 0.245682i) q^{36} +(-0.406764 - 1.78215i) q^{37} +(-1.18253 - 5.18100i) q^{38} +(2.20346 + 2.76305i) q^{39} +(-0.265643 + 1.16386i) q^{40} -8.32895 q^{41} +(-0.745242 + 3.26512i) q^{42} +(3.31774 - 1.59774i) q^{43} +(-3.31774 + 1.59774i) q^{44} +(0.0834753 - 0.365729i) q^{45} -5.89351 q^{46} +(0.220911 - 0.967876i) q^{47} +(1.02179 + 1.28129i) q^{48} +(0.628344 + 2.75296i) q^{49} +(0.795484 + 3.48524i) q^{50} +(-6.67760 + 8.37344i) q^{51} +(-1.94290 + 0.935653i) q^{52} +(5.10353 + 2.45773i) q^{53} +(-3.38647 - 4.24650i) q^{54} +(2.74087 - 3.43695i) q^{55} +(-1.84121 - 0.886679i) q^{56} +8.70913 q^{57} +(-3.78671 + 3.82894i) q^{58} +2.94918 q^{59} +(-1.76267 - 0.848856i) q^{60} +(-1.12786 + 1.41429i) q^{61} +(5.60932 + 7.03387i) q^{62} +(0.578579 + 0.278629i) q^{63} +(-0.900969 + 0.433884i) q^{64} +(1.60508 - 2.01271i) q^{65} +(-1.34288 - 5.88354i) q^{66} +(-1.34482 - 5.89206i) q^{67} +(-4.07461 - 5.10940i) q^{68} +(2.14921 - 9.41630i) q^{69} +2.43961 q^{70} +(-0.836003 + 3.66277i) q^{71} +(0.283120 - 0.136343i) q^{72} +(-11.4647 + 5.52110i) q^{73} +(-0.406764 + 1.78215i) q^{74} -5.85860 q^{75} +(-1.18253 + 5.18100i) q^{76} +(4.69197 + 5.88354i) q^{77} +(-0.786405 - 3.44546i) q^{78} +(-3.14487 - 13.7786i) q^{79} +(0.744314 - 0.933340i) q^{80} +(7.17039 - 3.45308i) q^{81} +(7.50412 + 3.61380i) q^{82} +(1.27960 + 1.60457i) q^{83} +(2.08812 - 2.61842i) q^{84} +(7.02900 + 3.38499i) q^{85} -3.68242 q^{86} +(-4.73674 - 7.44650i) q^{87} +3.68242 q^{88} +(14.8768 + 7.16429i) q^{89} +(-0.233893 + 0.293292i) q^{90} +(2.74767 + 3.44546i) q^{91} +(5.30987 + 2.55710i) q^{92} +(-13.2839 + 6.39717i) q^{93} +(-0.618980 + 0.776176i) q^{94} +(-1.41169 - 6.18501i) q^{95} +(-0.364674 - 1.59774i) q^{96} +(-2.86572 - 3.59350i) q^{97} +(0.628344 - 2.75296i) q^{98} -1.15716 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
\(3\) 1.02179 − 1.28129i 0.589933 − 0.739752i −0.393839 − 0.919180i \(-0.628853\pi\)
0.983771 + 0.179428i \(0.0574245\pi\)
\(4\) 0.623490 + 0.781831i 0.311745 + 0.390916i
\(5\) −1.07557 − 0.517965i −0.481007 − 0.231641i 0.177636 − 0.984096i \(-0.443155\pi\)
−0.658643 + 0.752455i \(0.728869\pi\)
\(6\) −1.47653 + 0.711061i −0.602793 + 0.290290i
\(7\) 1.27416 − 1.59774i 0.481585 − 0.603889i −0.480380 − 0.877061i \(-0.659501\pi\)
0.961965 + 0.273171i \(0.0880726\pi\)
\(8\) −0.222521 − 0.974928i −0.0786730 − 0.344689i
\(9\) 0.0699247 + 0.306360i 0.0233082 + 0.102120i
\(10\) 0.744314 + 0.933340i 0.235373 + 0.295148i
\(11\) −0.819415 + 3.59009i −0.247063 + 1.08245i 0.687368 + 0.726309i \(0.258766\pi\)
−0.934431 + 0.356144i \(0.884091\pi\)
\(12\) 1.63883 0.473089
\(13\) −0.479858 + 2.10239i −0.133089 + 0.583099i 0.863769 + 0.503888i \(0.168097\pi\)
−0.996858 + 0.0792116i \(0.974760\pi\)
\(14\) −1.84121 + 0.886679i −0.492083 + 0.236975i
\(15\) −1.76267 + 0.848856i −0.455119 + 0.219174i
\(16\) −0.222521 + 0.974928i −0.0556302 + 0.243732i
\(17\) −6.53517 −1.58501 −0.792506 − 0.609864i \(-0.791224\pi\)
−0.792506 + 0.609864i \(0.791224\pi\)
\(18\) 0.0699247 − 0.306360i 0.0164814 − 0.0722098i
\(19\) 3.31337 + 4.15484i 0.760140 + 0.953186i 0.999844 − 0.0176605i \(-0.00562181\pi\)
−0.239704 + 0.970846i \(0.577050\pi\)
\(20\) −0.265643 − 1.16386i −0.0593995 − 0.260246i
\(21\) −0.745242 − 3.26512i −0.162625 − 0.712508i
\(22\) 2.29595 − 2.87903i 0.489498 − 0.613811i
\(23\) 5.30987 − 2.55710i 1.10718 − 0.533192i 0.211274 − 0.977427i \(-0.432239\pi\)
0.895910 + 0.444235i \(0.146524\pi\)
\(24\) −1.47653 − 0.711061i −0.301396 − 0.145145i
\(25\) −2.22890 − 2.79495i −0.445779 − 0.558989i
\(26\) 1.34453 − 1.68599i 0.263684 − 0.330650i
\(27\) 4.89359 + 2.35663i 0.941771 + 0.453533i
\(28\) 2.04359 0.386202
\(29\) 1.75040 − 5.09275i 0.325040 − 0.945700i
\(30\) 1.95641 0.357191
\(31\) −8.10571 − 3.90350i −1.45583 − 0.701090i −0.472232 − 0.881475i \(-0.656551\pi\)
−0.983596 + 0.180385i \(0.942266\pi\)
\(32\) 0.623490 − 0.781831i 0.110218 − 0.138210i
\(33\) 3.76267 + 4.71824i 0.654996 + 0.821339i
\(34\) 5.88799 + 2.83551i 1.00978 + 0.486285i
\(35\) −2.19801 + 1.05851i −0.371532 + 0.178920i
\(36\) −0.195925 + 0.245682i −0.0326541 + 0.0409470i
\(37\) −0.406764 − 1.78215i −0.0668717 − 0.292984i 0.930423 − 0.366487i \(-0.119439\pi\)
−0.997295 + 0.0735026i \(0.976582\pi\)
\(38\) −1.18253 − 5.18100i −0.191832 − 0.840469i
\(39\) 2.20346 + 2.76305i 0.352836 + 0.442442i
\(40\) −0.265643 + 1.16386i −0.0420018 + 0.184022i
\(41\) −8.32895 −1.30076 −0.650382 − 0.759608i \(-0.725391\pi\)
−0.650382 + 0.759608i \(0.725391\pi\)
\(42\) −0.745242 + 3.26512i −0.114993 + 0.503819i
\(43\) 3.31774 − 1.59774i 0.505951 − 0.243653i −0.163454 − 0.986551i \(-0.552263\pi\)
0.669405 + 0.742898i \(0.266549\pi\)
\(44\) −3.31774 + 1.59774i −0.500168 + 0.240868i
\(45\) 0.0834753 − 0.365729i 0.0124438 − 0.0545197i
\(46\) −5.89351 −0.868951
\(47\) 0.220911 − 0.967876i 0.0322232 − 0.141179i −0.956257 − 0.292526i \(-0.905504\pi\)
0.988481 + 0.151347i \(0.0483612\pi\)
\(48\) 1.02179 + 1.28129i 0.147483 + 0.184938i
\(49\) 0.628344 + 2.75296i 0.0897635 + 0.393280i
\(50\) 0.795484 + 3.48524i 0.112498 + 0.492888i
\(51\) −6.67760 + 8.37344i −0.935050 + 1.17252i
\(52\) −1.94290 + 0.935653i −0.269432 + 0.129752i
\(53\) 5.10353 + 2.45773i 0.701023 + 0.337595i 0.750215 − 0.661194i \(-0.229950\pi\)
−0.0491913 + 0.998789i \(0.515664\pi\)
\(54\) −3.38647 − 4.24650i −0.460840 − 0.577875i
\(55\) 2.74087 − 3.43695i 0.369579 − 0.463438i
\(56\) −1.84121 − 0.886679i −0.246042 − 0.118487i
\(57\) 8.70913 1.15355
\(58\) −3.78671 + 3.82894i −0.497220 + 0.502765i
\(59\) 2.94918 0.383951 0.191975 − 0.981400i \(-0.438511\pi\)
0.191975 + 0.981400i \(0.438511\pi\)
\(60\) −1.76267 − 0.848856i −0.227559 − 0.109587i
\(61\) −1.12786 + 1.41429i −0.144407 + 0.181081i −0.848775 − 0.528754i \(-0.822659\pi\)
0.704368 + 0.709835i \(0.251231\pi\)
\(62\) 5.60932 + 7.03387i 0.712385 + 0.893302i
\(63\) 0.578579 + 0.278629i 0.0728941 + 0.0351040i
\(64\) −0.900969 + 0.433884i −0.112621 + 0.0542355i
\(65\) 1.60508 − 2.01271i 0.199086 − 0.249646i
\(66\) −1.34288 − 5.88354i −0.165297 − 0.724214i
\(67\) −1.34482 − 5.89206i −0.164296 − 0.719830i −0.988209 − 0.153113i \(-0.951070\pi\)
0.823912 − 0.566717i \(-0.191787\pi\)
\(68\) −4.07461 − 5.10940i −0.494120 − 0.619606i
\(69\) 2.14921 − 9.41630i 0.258734 − 1.13359i
\(70\) 2.43961 0.291589
\(71\) −0.836003 + 3.66277i −0.0992153 + 0.434691i 0.900785 + 0.434266i \(0.142992\pi\)
−1.00000 0.000424562i \(0.999865\pi\)
\(72\) 0.283120 − 0.136343i 0.0333660 − 0.0160682i
\(73\) −11.4647 + 5.52110i −1.34184 + 0.646196i −0.960511 − 0.278243i \(-0.910248\pi\)
−0.381329 + 0.924439i \(0.624534\pi\)
\(74\) −0.406764 + 1.78215i −0.0472854 + 0.207171i
\(75\) −5.85860 −0.676493
\(76\) −1.18253 + 5.18100i −0.135645 + 0.594301i
\(77\) 4.69197 + 5.88354i 0.534700 + 0.670492i
\(78\) −0.786405 − 3.44546i −0.0890428 − 0.390122i
\(79\) −3.14487 − 13.7786i −0.353826 − 1.55021i −0.768264 − 0.640133i \(-0.778879\pi\)
0.414438 − 0.910077i \(-0.363978\pi\)
\(80\) 0.744314 − 0.933340i 0.0832169 − 0.104351i
\(81\) 7.17039 − 3.45308i 0.796710 − 0.383676i
\(82\) 7.50412 + 3.61380i 0.828692 + 0.399077i
\(83\) 1.27960 + 1.60457i 0.140455 + 0.176125i 0.847083 − 0.531460i \(-0.178356\pi\)
−0.706629 + 0.707584i \(0.749785\pi\)
\(84\) 2.08812 − 2.61842i 0.227833 − 0.285693i
\(85\) 7.02900 + 3.38499i 0.762403 + 0.367154i
\(86\) −3.68242 −0.397085
\(87\) −4.73674 − 7.44650i −0.507832 − 0.798349i
\(88\) 3.68242 0.392547
\(89\) 14.8768 + 7.16429i 1.57694 + 0.759413i 0.998416 − 0.0562548i \(-0.0179159\pi\)
0.578521 + 0.815668i \(0.303630\pi\)
\(90\) −0.233893 + 0.293292i −0.0246544 + 0.0309157i
\(91\) 2.74767 + 3.44546i 0.288034 + 0.361183i
\(92\) 5.30987 + 2.55710i 0.553592 + 0.266596i
\(93\) −13.2839 + 6.39717i −1.37747 + 0.663356i
\(94\) −0.618980 + 0.776176i −0.0638429 + 0.0800565i
\(95\) −1.41169 − 6.18501i −0.144836 − 0.634569i
\(96\) −0.364674 − 1.59774i −0.0372194 − 0.163069i
\(97\) −2.86572 − 3.59350i −0.290970 − 0.364864i 0.614764 − 0.788711i \(-0.289251\pi\)
−0.905734 + 0.423846i \(0.860680\pi\)
\(98\) 0.628344 − 2.75296i 0.0634724 − 0.278091i
\(99\) −1.15716 −0.116299
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.45.2 ✓ 12
3.2 odd 2 522.2.k.h.451.2 12
4.3 odd 2 464.2.u.h.161.1 12
29.3 odd 28 1682.2.b.i.1681.3 12
29.7 even 7 1682.2.a.t.1.4 6
29.20 even 7 inner 58.2.d.b.49.2 yes 12
29.22 even 14 1682.2.a.q.1.3 6
29.26 odd 28 1682.2.b.i.1681.10 12
87.20 odd 14 522.2.k.h.397.2 12
116.107 odd 14 464.2.u.h.49.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.45.2 ✓ 12 1.1 even 1 trivial
58.2.d.b.49.2 yes 12 29.20 even 7 inner
464.2.u.h.49.1 12 116.107 odd 14
464.2.u.h.161.1 12 4.3 odd 2
522.2.k.h.397.2 12 87.20 odd 14
522.2.k.h.451.2 12 3.2 odd 2
1682.2.a.q.1.3 6 29.22 even 14
1682.2.a.t.1.4 6 29.7 even 7
1682.2.b.i.1681.3 12 29.3 odd 28
1682.2.b.i.1681.10 12 29.26 odd 28