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Properties

Label 58.2.d.b.53.1
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 53.1
Root \(0.760453 - 3.33176i\) of defining polynomial
Character \(\chi\) \(=\) 58.53
Dual form 58.2.d.b.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} +(-0.760453 - 3.33176i) q^{3} +(-0.222521 + 0.974928i) q^{4} +(1.00725 + 1.26305i) q^{5} +(2.13074 - 2.67187i) q^{6} +(0.338433 + 1.48277i) q^{7} +(-0.900969 + 0.433884i) q^{8} +(-7.81944 + 3.76565i) q^{9} +(-0.359484 + 1.57500i) q^{10} +(-1.70872 - 0.822877i) q^{11} +3.41744 q^{12} +(2.87014 + 1.38219i) q^{13} +(-0.948269 + 1.18909i) q^{14} +(3.44223 - 4.31642i) q^{15} +(-0.900969 - 0.433884i) q^{16} -1.69929 q^{17} +(-7.81944 - 3.76565i) q^{18} +(0.818639 - 3.58669i) q^{19} +(-1.45552 + 0.700942i) q^{20} +(4.68289 - 2.25516i) q^{21} +(-0.422020 - 1.84899i) q^{22} +(-1.67782 + 2.10392i) q^{23} +(2.13074 + 2.67187i) q^{24} +(0.531856 - 2.33021i) q^{25} +(0.708867 + 3.10575i) q^{26} +(12.1003 + 15.1733i) q^{27} -1.52091 q^{28} +(-4.31703 - 3.21920i) q^{29} +5.52091 q^{30} +(-1.11130 - 1.39353i) q^{31} +(-0.222521 - 0.974928i) q^{32} +(-1.44223 + 6.31882i) q^{33} +(-1.05949 - 1.32856i) q^{34} +(-1.53194 + 1.92099i) q^{35} +(-1.93124 - 8.46133i) q^{36} +(6.37006 - 3.06766i) q^{37} +(3.31460 - 1.59623i) q^{38} +(2.42251 - 10.6137i) q^{39} +(-1.45552 - 0.700942i) q^{40} -5.03259 q^{41} +(4.68289 + 2.25516i) q^{42} +(-1.18247 + 1.48277i) q^{43} +(1.18247 - 1.48277i) q^{44} +(-12.6324 - 6.08343i) q^{45} -2.69101 q^{46} +(-2.31606 - 1.11536i) q^{47} +(-0.760453 + 3.33176i) q^{48} +(4.22270 - 2.03355i) q^{49} +(2.15344 - 1.03704i) q^{50} +(1.29223 + 5.66162i) q^{51} +(-1.98620 + 2.49062i) q^{52} +(5.14319 + 6.44936i) q^{53} +(-4.31856 + 18.9208i) q^{54} +(-0.681776 - 2.98705i) q^{55} +(-0.948269 - 1.18909i) q^{56} -12.5725 q^{57} +(-0.174748 - 5.38233i) q^{58} +2.95027 q^{59} +(3.44223 + 4.31642i) q^{60} +(1.72363 + 7.55173i) q^{61} +(0.396619 - 1.73770i) q^{62} +(-8.22996 - 10.3200i) q^{63} +(0.623490 - 0.781831i) q^{64} +(1.14518 + 5.01736i) q^{65} +(-5.83946 + 2.81214i) q^{66} +(-2.23359 + 1.07564i) q^{67} +(0.378127 - 1.65668i) q^{68} +(8.28565 + 3.99016i) q^{69} -2.45703 q^{70} +(-5.46771 - 2.63311i) q^{71} +(5.41122 - 6.78546i) q^{72} +(2.33692 - 2.93041i) q^{73} +(6.37006 + 3.06766i) q^{74} -8.16816 q^{75} +(3.31460 + 1.59623i) q^{76} +(0.641852 - 2.81214i) q^{77} +(9.80856 - 4.72355i) q^{78} +(-8.30753 + 4.00069i) q^{79} +(-0.359484 - 1.57500i) q^{80} +(25.1185 - 31.4977i) q^{81} +(-3.13777 - 3.93464i) q^{82} +(1.22901 - 5.38465i) q^{83} +(1.15658 + 5.06730i) q^{84} +(-1.71161 - 2.14629i) q^{85} -1.89654 q^{86} +(-7.44272 + 16.8314i) q^{87} +1.89654 q^{88} +(9.95740 + 12.4862i) q^{89} +(-3.11994 - 13.6693i) q^{90} +(-1.07812 + 4.72355i) q^{91} +(-1.67782 - 2.10392i) q^{92} +(-3.79781 + 4.76231i) q^{93} +(-0.572020 - 2.50618i) q^{94} +(5.35476 - 2.57872i) q^{95} +(-3.07901 + 1.48277i) q^{96} +(-3.31730 + 14.5341i) q^{97} +(4.22270 + 2.03355i) q^{98} +16.4599 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
\(3\) −0.760453 − 3.33176i −0.439048 − 1.92359i −0.378995 − 0.925399i \(-0.623730\pi\)
−0.0600525 − 0.998195i \(-0.519127\pi\)
\(4\) −0.222521 + 0.974928i −0.111260 + 0.487464i
\(5\) 1.00725 + 1.26305i 0.450457 + 0.564855i 0.954266 − 0.298960i \(-0.0966398\pi\)
−0.503809 + 0.863815i \(0.668068\pi\)
\(6\) 2.13074 − 2.67187i 0.869872 − 1.09078i
\(7\) 0.338433 + 1.48277i 0.127916 + 0.560436i 0.997747 + 0.0670866i \(0.0213704\pi\)
−0.869831 + 0.493349i \(0.835772\pi\)
\(8\) −0.900969 + 0.433884i −0.318541 + 0.153401i
\(9\) −7.81944 + 3.76565i −2.60648 + 1.25522i
\(10\) −0.359484 + 1.57500i −0.113679 + 0.498060i
\(11\) −1.70872 − 0.822877i −0.515199 − 0.248107i 0.158172 − 0.987412i \(-0.449440\pi\)
−0.673371 + 0.739305i \(0.735154\pi\)
\(12\) 3.41744 0.986531
\(13\) 2.87014 + 1.38219i 0.796035 + 0.383350i 0.787268 − 0.616611i \(-0.211495\pi\)
0.00876713 + 0.999962i \(0.497209\pi\)
\(14\) −0.948269 + 1.18909i −0.253436 + 0.317798i
\(15\) 3.44223 − 4.31642i 0.888780 − 1.11449i
\(16\) −0.900969 − 0.433884i −0.225242 − 0.108471i
\(17\) −1.69929 −0.412138 −0.206069 − 0.978537i \(-0.566067\pi\)
−0.206069 + 0.978537i \(0.566067\pi\)
\(18\) −7.81944 − 3.76565i −1.84306 − 0.887571i
\(19\) 0.818639 − 3.58669i 0.187809 − 0.822843i −0.789960 − 0.613158i \(-0.789899\pi\)
0.977769 − 0.209685i \(-0.0672439\pi\)
\(20\) −1.45552 + 0.700942i −0.325465 + 0.156735i
\(21\) 4.68289 − 2.25516i 1.02189 − 0.492116i
\(22\) −0.422020 − 1.84899i −0.0899749 − 0.394206i
\(23\) −1.67782 + 2.10392i −0.349849 + 0.438697i −0.925356 − 0.379100i \(-0.876234\pi\)
0.575506 + 0.817797i \(0.304805\pi\)
\(24\) 2.13074 + 2.67187i 0.434936 + 0.545392i
\(25\) 0.531856 − 2.33021i 0.106371 − 0.466042i
\(26\) 0.708867 + 3.10575i 0.139020 + 0.609088i
\(27\) 12.1003 + 15.1733i 2.32871 + 2.92011i
\(28\) −1.52091 −0.287424
\(29\) −4.31703 − 3.21920i −0.801652 − 0.597791i
\(30\) 5.52091 1.00797
\(31\) −1.11130 − 1.39353i −0.199596 − 0.250285i 0.671953 − 0.740594i \(-0.265456\pi\)
−0.871549 + 0.490308i \(0.836884\pi\)
\(32\) −0.222521 − 0.974928i −0.0393365 − 0.172345i
\(33\) −1.44223 + 6.31882i −0.251060 + 1.09996i
\(34\) −1.05949 − 1.32856i −0.181701 − 0.227846i
\(35\) −1.53194 + 1.92099i −0.258944 + 0.324706i
\(36\) −1.93124 − 8.46133i −0.321874 − 1.41022i
\(37\) 6.37006 − 3.06766i 1.04723 − 0.504320i 0.170529 − 0.985353i \(-0.445452\pi\)
0.876703 + 0.481032i \(0.159738\pi\)
\(38\) 3.31460 − 1.59623i 0.537699 − 0.258942i
\(39\) 2.42251 − 10.6137i 0.387913 − 1.69956i
\(40\) −1.45552 − 0.700942i −0.230138 − 0.110829i
\(41\) −5.03259 −0.785959 −0.392979 − 0.919547i \(-0.628556\pi\)
−0.392979 + 0.919547i \(0.628556\pi\)
\(42\) 4.68289 + 2.25516i 0.722585 + 0.347979i
\(43\) −1.18247 + 1.48277i −0.180325 + 0.226121i −0.863776 − 0.503876i \(-0.831907\pi\)
0.683451 + 0.729997i \(0.260478\pi\)
\(44\) 1.18247 − 1.48277i 0.178264 − 0.223537i
\(45\) −12.6324 − 6.08343i −1.88312 − 0.906864i
\(46\) −2.69101 −0.396768
\(47\) −2.31606 − 1.11536i −0.337832 − 0.162691i 0.257274 − 0.966339i \(-0.417176\pi\)
−0.595106 + 0.803647i \(0.702890\pi\)
\(48\) −0.760453 + 3.33176i −0.109762 + 0.480898i
\(49\) 4.22270 − 2.03355i 0.603243 − 0.290507i
\(50\) 2.15344 − 1.03704i 0.304542 − 0.146660i
\(51\) 1.29223 + 5.66162i 0.180948 + 0.792786i
\(52\) −1.98620 + 2.49062i −0.275437 + 0.345387i
\(53\) 5.14319 + 6.44936i 0.706472 + 0.885888i 0.997488 − 0.0708303i \(-0.0225649\pi\)
−0.291016 + 0.956718i \(0.593993\pi\)
\(54\) −4.31856 + 18.9208i −0.587681 + 2.57480i
\(55\) −0.681776 − 2.98705i −0.0919306 − 0.402774i
\(56\) −0.948269 − 1.18909i −0.126718 − 0.158899i
\(57\) −12.5725 −1.66527
\(58\) −0.174748 − 5.38233i −0.0229456 − 0.706734i
\(59\) 2.95027 0.384093 0.192046 − 0.981386i \(-0.438488\pi\)
0.192046 + 0.981386i \(0.438488\pi\)
\(60\) 3.44223 + 4.31642i 0.444390 + 0.557247i
\(61\) 1.72363 + 7.55173i 0.220688 + 0.966899i 0.956962 + 0.290215i \(0.0937268\pi\)
−0.736273 + 0.676685i \(0.763416\pi\)
\(62\) 0.396619 − 1.73770i 0.0503707 − 0.220688i
\(63\) −8.22996 − 10.3200i −1.03688 − 1.30020i
\(64\) 0.623490 − 0.781831i 0.0779362 − 0.0977289i
\(65\) 1.14518 + 5.01736i 0.142042 + 0.622327i
\(66\) −5.83946 + 2.81214i −0.718788 + 0.346150i
\(67\) −2.23359 + 1.07564i −0.272876 + 0.131410i −0.565318 − 0.824873i \(-0.691246\pi\)
0.292442 + 0.956283i \(0.405532\pi\)
\(68\) 0.378127 − 1.65668i 0.0458546 − 0.200902i
\(69\) 8.28565 + 3.99016i 0.997475 + 0.480359i
\(70\) −2.45703 −0.293672
\(71\) −5.46771 − 2.63311i −0.648898 − 0.312493i 0.0803250 − 0.996769i \(-0.474404\pi\)
−0.729223 + 0.684276i \(0.760118\pi\)
\(72\) 5.41122 − 6.78546i 0.637719 − 0.799674i
\(73\) 2.33692 − 2.93041i 0.273516 − 0.342978i −0.626034 − 0.779796i \(-0.715323\pi\)
0.899550 + 0.436817i \(0.143894\pi\)
\(74\) 6.37006 + 3.06766i 0.740505 + 0.356608i
\(75\) −8.16816 −0.943178
\(76\) 3.31460 + 1.59623i 0.380211 + 0.183100i
\(77\) 0.641852 − 2.81214i 0.0731458 − 0.320473i
\(78\) 9.80856 − 4.72355i 1.11060 − 0.534837i
\(79\) −8.30753 + 4.00069i −0.934670 + 0.450113i −0.838286 − 0.545231i \(-0.816442\pi\)
−0.0963841 + 0.995344i \(0.530728\pi\)
\(80\) −0.359484 − 1.57500i −0.0401915 − 0.176091i
\(81\) 25.1185 − 31.4977i 2.79095 − 3.49974i
\(82\) −3.13777 − 3.93464i −0.346509 − 0.434508i
\(83\) 1.22901 − 5.38465i 0.134902 − 0.591042i −0.861609 − 0.507573i \(-0.830543\pi\)
0.996510 − 0.0834694i \(-0.0266001\pi\)
\(84\) 1.15658 + 5.06730i 0.126193 + 0.552887i
\(85\) −1.71161 − 2.14629i −0.185650 − 0.232798i
\(86\) −1.89654 −0.204509
\(87\) −7.44272 + 16.8314i −0.797944 + 1.80451i
\(88\) 1.89654 0.202172
\(89\) 9.95740 + 12.4862i 1.05548 + 1.32353i 0.944065 + 0.329759i \(0.106968\pi\)
0.111417 + 0.993774i \(0.464461\pi\)
\(90\) −3.11994 − 13.6693i −0.328870 − 1.44087i
\(91\) −1.07812 + 4.72355i −0.113018 + 0.495163i
\(92\) −1.67782 − 2.10392i −0.174925 − 0.219348i
\(93\) −3.79781 + 4.76231i −0.393815 + 0.493828i
\(94\) −0.572020 − 2.50618i −0.0589994 − 0.258493i
\(95\) 5.35476 − 2.57872i 0.549387 − 0.264571i
\(96\) −3.07901 + 1.48277i −0.314250 + 0.151335i
\(97\) −3.31730 + 14.5341i −0.336821 + 1.47571i 0.468814 + 0.883297i \(0.344682\pi\)
−0.805635 + 0.592412i \(0.798176\pi\)
\(98\) 4.22270 + 2.03355i 0.426557 + 0.205419i
\(99\) 16.4599 1.65428
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.53.1 yes 12
3.2 odd 2 522.2.k.h.343.1 12
4.3 odd 2 464.2.u.h.401.2 12
29.8 odd 28 1682.2.b.i.1681.12 12
29.9 even 14 1682.2.a.q.1.1 6
29.20 even 7 1682.2.a.t.1.6 6
29.21 odd 28 1682.2.b.i.1681.1 12
29.23 even 7 inner 58.2.d.b.23.1 ✓ 12
87.23 odd 14 522.2.k.h.487.1 12
116.23 odd 14 464.2.u.h.81.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.d.b.23.1 ✓ 12 29.23 even 7 inner
58.2.d.b.53.1 yes 12 1.1 even 1 trivial
464.2.u.h.81.2 12 116.23 odd 14
464.2.u.h.401.2 12 4.3 odd 2
522.2.k.h.343.1 12 3.2 odd 2
522.2.k.h.487.1 12 87.23 odd 14
1682.2.a.q.1.1 6 29.9 even 14
1682.2.a.t.1.6 6 29.20 even 7
1682.2.b.i.1681.1 12 29.21 odd 28
1682.2.b.i.1681.12 12 29.8 odd 28